Ngoc Hoang

Minh

On a conjecture by Pierre Cartier about a group of associators

, Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0, x1} with coefficients in a Q-extension, A, subjected to some suitable conditions, there exists an unique algebra homomorphism ϕ from the Q-algebra generated by the convergent polyzêtas to A such that Φ is computed from ΦKZ Drinfel'd associator by applying ϕ to each coefficient. We prove ϕ exists and it is a free Lie exponential over X. Moreover, we give a complete description of the kernel of polyzêta and draw some consequences about a structure of the algebra of convergent polyzêtas and about the arithmetical nature of the Euler constant.

In 1986, in order to study the linear representation of the braid group B n coming from the monodromy of the Knizhnik-Zamolodchikov differential equations over C n * = {z = (z 1 , . . . , z n ) ∈ C n | z i = z j for i = j} [START_REF] Drinfel | d.-Quantum group[END_REF] :

dF (z) = Ω n (z)F (z) with Ω n (z) = 1 2iπ 1≤i<j≤n t i,j d(z i -z j ) z i -z j , (1)
and {t i,j } i,j≥1 are noncommutative variables, Drinfel'd introduced a class of formal power series Φ on noncommutative variables over the finite alphabet X = {x 0 , x 1 }. Such a power series Φ is called an associator.

Since the system (1) is completely integrable then dΩ n -Ω n ∧ Ω n = 0 [START_REF] Cartier | Développements récents sur les groupes de tresses. Applications à la topologie et à l'algèbre[END_REF][START_REF] Drinfel | d.-Quantum group[END_REF]. It is equivalent to the fact the {t i,j } i,j≥1 satisfy the infinitesimal braid relations :

t i,j = 0 for i = j, (2)
t i,j = t j,i for i = j, (3) [t i,j , t i,k + t j,k ] = 0 for distinct i, j, k,

[t i,j , t k,l ] = 0 for distinct i, j, k, l.

Example 1.

• T 2 = {t 1,2 }.

Ω 2 (z 1 , z 2 ) = t 1,2 2iπ d(z 1 -z 2 ) z 1 -z 2 with F (z 1 , z 2 ) = (z 1 -z 2 ) t1,2/2iπ . • T 3 = {t 1,2 , t 1,3 , t 2,3 }, [t 1,3 , t 1,2 + t 2,3 ] = [t 2,3 , t 1,2 + t 1,3 ] = 0. Ω 3 (z 1 , z 2 , z 3 ) = 1 2iπ t 1,2 d(z 1 -z 2 ) z 1 -z 2 + t 1,3 d(z 1 -z 3 ) z 1 -z 3 + t 2,3 d(z 2 -z 3 ) z 2 -z 3 . F (z 1 , z 2 , z 3 ) = G z 1 -z 2 z 1 -z 3 (z 1 -z 3 ) (t1,2+t1,3+t2,3)/2iπ ,
where G satisfies the following fuchsian differential equation with three regular singularities at 0, 1 and ∞ :

(DE) dG(z) = [x 0 ω 0 (z) + x 1 ω 1 (z)]G(z),
He also proved there exists the associator Φ KZ such that G -1 1 (z)G 0 (z) = Φ KZ . After that, Lê and Murakami expressed the coefficients of the Drinfel'd associator Φ KZ in terms of convergent polyzêtas [START_REF] Lê | Kontsevich's integral for Kauffman polynomial[END_REF], i.e. for r 1 > 1,

ζ(r 1 , . . . , r k ) = n1>...>n k >0 1 n r1 1 . . . n r k k . (7)
In [START_REF] Lê | Kontsevich's integral for Kauffman polynomial[END_REF], the authors also expressed the divergent coefficients as linear combinations of convergent polyzêtas via a regularization process (see also [START_REF] Ngoc | Differential Galois groups and noncommutative generating series of polylogarithms[END_REF]). This process is one of many ways to regularize the divergent terms.

Group of associators and regularized Chen generating series

The algebraic aspects of our regularization process based essentially on various products 1 among polyzêtas (see [START_REF] Hoang | De l'algèbre des ζ de Riemann multivariées l'algèbre des ζ de Hurwitz multivariées[END_REF]) and its analytical aspects will be described, in Section 3.1, as the finite part, of the asymptotic expansions in different scales of comparison 2 [5]. It will be seen also, in Section 3.2, as the action of the differential Galois group of the polylogarithms 3 (recalled in Section 2.

This action leads then to a conjecture by Pierre Cartier ([8], conjecture C3) and to the description of the group of associators yielding the ideal of polynomial relations among coefficients of associators (theorems 13 and 14). This group is in fact, closely linked to the group of the Chen generating series studied by K.T. Chen to describe the solutions of differential equations [START_REF] Chen | Iterated path integrals[END_REF] and it turns out that each associator regularizes a Chen generating series of the differential forms ω 0 and ω 1 along the integration path on the simply connected domain C -(] -∞, 0] ∪ [1, +∞[).

Global renormalization and global regularization

In fact, our regularization process based essentially on two noncommutative generating series over Y = {y i } i≥1 , which encodes the multi-indices (r 1 , . . . , r k ) by the words y r1 . . . y r k over the monoid generated by Y , denoted by Y * , of polylogarithms and of harmonic sums (recalled in Section 2.2.1)

Λ(z) =
Through the algebraic combinatorial aspects 4 [START_REF] Reutenauer | Free Lie Algebras[END_REF] and the topological aspects [START_REF] Berstel | Rational series and their languages[END_REF] of formal power series in noncommutative variables, we have already 1 First source of ambiguity leading to the problem of rewriting expressions of polyzêtas in a canonical form using irreducible Lyndon words (see [START_REF] Ngoc | Lyndon words, polylogarithmic functions and the Riemann ζ function[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF]). 2 Second source of ambiguity leading to the problem to determine the value of regularized polyzêtas and its analytical meaning (see [START_REF] Hoang | De l'algèbre des ζ de Riemann multivariées l'algèbre des ζ de Hurwitz multivariées[END_REF][START_REF] Ngoc | Finite polyzêtas, Poly-Bernoulli numbers, identities of polyzêtas and noncommutative rational power series[END_REF]). 3 Third source of ambiguity leading to the problem of fixing the integration path to solve (DE) and its monodromy group (see [START_REF] Hoang Ngoc Minh | Polylogarithms and Shuffle Algebra[END_REF]) or its differential Galois group (see [START_REF] Ngoc | Differential Galois groups and noncommutative generating series of polylogarithms[END_REF]). 4 See [START_REF] Reutenauer | Free Lie Algebras[END_REF] to get an idea of these aspects of combinatorial Hopf algebra of the shuffle product,

showed the existence of noncommutative formal series over Y , Z 1 and Z 2 with constant terms, such that [START_REF] Ngoc | Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Inputs[END_REF] lim

z→1 exp y 1 log 1 1 -z Λ(z) = Z 1 , (11)
lim

N →∞ exp k≥1 H y k (N ) (-y 1 ) k k H(N ) = Z 2 . (12)
Moreover, Z 1 and Z 2 are equal and stand for the noncommutative generating series of all convergent polyzêtas {ζ(w)} w∈Y * -y1Y * as shown by the factorized form indexed by Lyndon words (recalled in Section 2.2). This theorem enables, in particular, to explicit the counter-terms eliminating the divergence of the polylogarithms {Li w (z)} w∈y1Y * , for z → 1, and of the harmonic sums {H w (N )} w∈y1Y * , for N → ∞, and to calculate the Euler-Mac Laurin constants associated to the divergent polyzêtas {ζ(w)} w∈y1Y * (see Corollary 4). It allows also to give, in Section 3.3 and via identification of locale coordinates in infinite dimension, a complete description of the kernel by its generators, of the following algebra homomorphism 5 ζ : (Aǫ ⊕ (Yy 1 )A Y , ) -→ (R, .)

y r1 . . . y r k -→ n1>...>n k >0 1 n r1 1 . . . n r k k , (13)
and the set of A-irreducible polyzêtas forming a transcendence basis of the image of ζ, with A = Q[iπ] (see Corollary 10). Finally, via the indiscernability (recalled in Section 2.3) over the group of associators, this study makes precise the structure of the A-algebra generated by the convergent polyzêtas (see Theorem 19) and concludes the main challenge of the polynomial relations among polyzêtas indexed by convergent Lyndon words which are algebraicly independant on the Euler constant and motivated [START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF][START_REF] Ngoc | Lyndon words, polylogarithmic functions and the Riemann ζ function[END_REF]3,[START_REF] Wardi | [END_REF]. In particular, the A-algebra generated by the convergent polyzêtas was conjectured to be free [START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF][START_REF] Ngoc | Lyndon words, polylogarithmic functions and the Riemann ζ function[END_REF] and it will be proved, thanks to the propositions 15, 16 and 17. Moreover, this free A-algebra is graded by weight meaning there is no linear relation among convergent polyzêtas of different weight (see Theorem 19).

denoted by ⊔⊔ , and its co-product, denoted by ∆ ⊔⊔ . For the quasi-shuffle product, denoted by , and its co-product, denoted by ∆ , see Annexe A. In our works, recalled in Annexe B, these algebraic combinatorial aspects were explored systematically to expand the outputs of nonlinear controlled dynamical system with singular inputs (Corollary 17) on polylogarithmic functional basis [START_REF] Ngoc | Input/Output behaviour of nonlinear control systems : about exact and approximated computations[END_REF][START_REF] Ngoc | Input/Output Behaviour of Nonlinear Control Systems : Rational Approximations, Nilpotent structural Approximations[END_REF][START_REF] Ngoc | Symbolic integration of meromorphic differential systems via Dirichlet functions[END_REF]. In this way [START_REF] Ngoc | Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Inputs[END_REF], polyzêtas do appear then as fundamental arithmetical constant for the asymptotic analysis and for the renormalization of the outputs and their successive derivations (Corollary 18) via the extended Fliess fundamental formula (Theorem 23). 5 Here, ǫ stands for the empty word over Y .

2 Background : structures and analytical studies of harmonic sums and of polylogarithms Let {t i } i∈N+ be an infinite set of variables. The elementary symmetric functions η k and the power sums ψ k are defined by (see [START_REF] Reutenauer | Free Lie Algebras[END_REF])

η k (t) = n1>...>n k >0 t n1 . . . t n k and ψ k (t) = n>0 t k n . (15)
They are respectively coefficients of the following generating functions

η(t | z) = i≥1 (1 + t i z) and ψ(t | z) = i≥1 t i z 1 -t i z . ( 16
)
These generating functions satisfy a Newton identity

z d dz log η(t | z) = ψ(t | -z). (17)
The fundamental theorem from symmetric functions theory asserts that {η k } k≥0 are linearly independent, and provides remarkable identities like (with η 0 = 1) :

η k = (-1) k k! s 1 ,...,s k ≥0 s 1 +...+ks k =k+1 k s 1 , . . . , s k - ψ 1 1 s1 . . . - ψ k k s k . (18)
Let Y be the infinite alphabet {y i } i≥1 equipped with the order y 1 > y 2 > y 3 > . . . and let LynY be the set of Lyndon words over Y . The length of w = y s1 . . . y sr ∈ Y * is denoted by | w | and its degree equals to s 1 + . . . + s r .

The quasi-symmetric function F w , of depth r =| w | and of degree (or weight) s 1 + . . . + s r , is defined by

F w (t) = n1>...>nr >0 t s1 n1 . . . t sr nr . (19)
In particular, F y k 1 = η k and F y k = ψ k . The functions {F y k 1 } k≥0 are linearly independent and integrating differential equation [START_REF] Gari | la dimorphie et l'arithmétique des multizêtas : un premier bilan[END_REF] shows that functions F y k 1 and F y k are linked by the formula

k≥0 F y k 1 z k = exp - k≥1 F y k (-z) k k . ( 20
)
Every H w (N ) can be obtained by specializing, in the quasi-symmetric function F w , the variables {t i } i≥1 as follows [START_REF] Hoffman | The algebra of multiple harmonic series[END_REF]

∀N ≥ i ≥ 1, t i = 1/i and ∀i > N, t i = 0. ( 21
)
In the same way, for w ∈ Y *y 1 Y * , the convergent polyzêta ζ(w) can be obtained by specializing, in F w , the variables {t i } i≥1 as follows [START_REF] Hoffman | The algebra of multiple harmonic series[END_REF] ∀N ≥ i ≥ 1,

t i = 1/i. ( 22
)
The notation F w is extended by linearity to all polynomials over Y . If u, v ∈ Y * , of length r, s and of weight 6 p, q respectively, F u v is a quasisymmetric function of depth r + s and of weight p + q, and F u v = F u F v , where is the quasi-shuffle product 7 [START_REF] Hoffman | The algebra of multiple harmonic series[END_REF]. Hence,

∀u, v ∈ Y * , H u v = H u H v , ( 23
) ⇒ ∀u, v ∈ Y * -y 1 Y * , ζ(u v) = ζ(u) ζ(v). (24)
Remarkable identity ( 18) can be then seen as

y k 1 = (-1) k k! s 1 ,...,s k ≥0 s 1 +...+ks k =k+1 k s 1 , . . . , s k (-y 1 ) s1 1 s1 . . . (-y k ) s k k s k . ( 25
)

Iterated integrals and polylogarithms

Let X be the finite alphabet {x 0 , x 1 } equipped with the order x 0 < x 1 8 . Let

C := C z, 1 z , 1 1 -z and G := z, 1 z , z -1 z , z z -1 , 1 1 -z , 1 -z . ( 26
)
This ring C is invariant under differentiation and under the homographic transformations belonging to the group G whose elements commute the singularities {0, 1, +∞}. The iterated integral over ω 0 , ω 1 associated to the word w = x i1 • • • x i k over X * (the monoid generated by X) and along the integration path z 0 z is the following multiple integral defined by [START_REF] Ngoc | Input/Output Behaviour of Nonlinear Control Systems : Rational Approximations, Nilpotent structural Approximations[END_REF] where t 1 • • • t r-1 is a subdivision of the path z 0 z. In a shortened notation, we denote this integral by α z z0 (w) and 9 α z z0 (ǫ) = 1. One can check that the polylogarithm Li s1,...,sr is also the value of the iterated integral over ω 0 , ω 1 and along the integration path 0 z [START_REF] Ngoc | Fonctions de Dirichlet d'ordre n et de paramètre t[END_REF][START_REF] Ngoc | Symbolic integration of meromorphic differential systems via Dirichlet functions[END_REF] :

z0 z ω i1 • • • ω i k = z z0 ω i1 (t 1 ) t1 z0 ω i2 (t 2 ) . . . tr-2 z0 ω ir (t r-1 ) tr-1 z0 ω ir (t r ),
Li w (z) = α z 0 (x s1-1 0 x 1 . . . x sr -1 0 x 1 ). ( 28
)
The definition of polylogarithms is extended over the words w ∈ X * by putting Li x0 (z) := log z. The {Li w } w∈X * are C-linearly independent [START_REF] Hoang Ngoc Minh | Polylogarithms and Shuffle Algebra[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF]. The

6
The weight is as in Equation [START_REF] Fliess | Fonctionnelles causales non linéaires et indéterminées non commutatives[END_REF]. 7 See Annexe A, for the study of the Hopf algebra of which is not included in [START_REF] Reutenauer | Free Lie Algebras[END_REF]. 8 In all the sequel, we follow the notations of [START_REF] Berstel | Rational series and their languages[END_REF][START_REF] Reutenauer | Free Lie Algebras[END_REF]. 9 Here, ǫ stands for the empty word over X.

functions P w (z) := (1z) -1 Li w (z), w ∈ X * , are also C-linearly independent. Since, for w ∈ Y * , P w is the ordinary generating function of {H w (N )} N ≥0 [START_REF] Ngoc | Finite polyzêtas, Poly-Bernoulli numbers, identities of polyzêtas and noncommutative rational power series[END_REF] :

P w (z) = N ≥0 H w (N ) z N (29)
then, as a consequence of the classical isomorphism between convergent Taylor series and their associated sums, the harmonic sums {H w } w∈Y * are also Clinearly independent. Firstly, ker P = {0} and ker H = {0}, and secondly, P is a morphism for the Hadamard product :

P u (z) ⊙ P v (z) = N ≥0 H u (N )H v (N )z N = N ≥0 H u v (N )z N = P u v (z). ( 30
)
Proposition 1 ( [START_REF] Ngoc | Finite polyzêtas, Poly-Bernoulli numbers, identities of polyzêtas and noncommutative rational power series[END_REF]). Extended by linearity, the following maps are isomorphism of algebras

P : (C Y , ) -→ (C{P w } w∈Y * , ⊙) , u -→ P u , H : (C Y , ) -→ (C{H w } w∈Y * , .) , u -→ H u = {H u (N )} N ≥0 .
Studying the equivalence between action of {(1z) l } l∈Z over {P w (z)} w∈Y * and that of {N k } k∈Z over {H w (N )} w∈Y * (see [START_REF] Costermans | Algorithmic and combinatoric aspects of multiple harmonic sums[END_REF]), we have Theorem 1 ([39]). The Hadamard C-algebra of {P w } w∈Y * can be identified with that of {P l } l∈LynY . In the same way, the algebra of harmonic sums {H w } w∈Y * with polynomial coefficients can be identified with that of {H l } l∈LynY .

By Identity [START_REF] Ngoc | Contribution au développement d'outils informatiques pour résoudre des problèmes d'automatique non linéaire[END_REF] and by applying the isomorphism H on the set of Lyndon words {y r } 1≤r≤k , we obtain H y k 1 as polynomials in {H yr } 1≤r≤k (which are algebraically independent), and

H y k 1 = s 1 ,...,s k ≥0 s 1 +...+ks k =k+1 (-1) k s 1 ! . . . s k ! - H y1 1 s1 . . . - H y k k s k . ( 31
)
2.2 Results à la Abel for generating series of harmonic sums and of polylogarithms

Generating series of harmonic sums and of polylogarithms

Let H(N ) be the noncommutative generating series of {H w (N )} w∈Y * [START_REF] Ngoc | Finite polyzêtas, Poly-Bernoulli numbers, identities of polyzêtas and noncommutative rational power series[END_REF] :

H(N ) := w∈Y * H w (N ) w. (32)
Let {Σ w } w∈Y * , { Σw } w∈Y * be respectively a PBW basis of the envelopping algebra U(Lie Q Y ) and the quasi-shuffle algebra (Q Y ,

) (viewed as a Qmodule) on duality such that {Σ l } l∈LynX , { Σl } l∈LynX are respectively a basis of Lie Q Y and a transcendence basis of (Q Y , ) (see Annexe A).

Theorem 2 (Factorization of H). Let

H reg (N ) := ց l∈LynY -{y1} e H Σl (N ) Σ l .
Then H(N ) = e Hy 1 (N ) y1 H reg (N ).

Proof. See Annexe A.

For l ∈ LynY -{y 1 }, the polynomial Σ l is a finite combination of words in Y *y 1 Y * . Then we can state the following Definition 1. We set Z := H reg (∞).

The noncommutative generating series of polylogarithms [START_REF] Hoang Ngoc Minh | Polylogarithms and Shuffle Algebra[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF] L :=

w∈X * Li w w (33) satisfies Drinfel'd's differential equation (DE) of Example 1 dL = (x 0 ω 0 + x 1 ω 1 )L (34)
with boundary condition [START_REF] Drinfel | -Quasi-Hopf Algebras[END_REF][START_REF] Drinfel'd | -On quasitriangular quasi-hopf algebra and a group closely connected with gal(q/q)[END_REF] L(ε) ε→0 + e x0 log ε .

This enables us to prove that L is the exponential of a Lie series 10 [35, 32]. Hence, Proposition 2 (Logarithm of L, [START_REF] Ngoc | Differential Galois groups and noncommutative generating series of polylogarithms[END_REF]). Let π 1 (w) is the following Lie series

π 1 (w) = k≥1 (-1) k-1 k u1,...,u k ∈X + w | u 1 ⊔⊔ . . . ⊔⊔ u k u 1 . . . u k . Then log L(z) = k≥1 (-1) k-1 k u1,...,u k ∈X + Li u1 ⊔⊔ ... ⊔⊔ u k (z) u 1 . . . u k = w∈X * Li w (z) π 1 (w).
Applying a theorem of Ree [START_REF] Ree | Lie elements and an algebra associated with shuffles[END_REF][START_REF] Reutenauer | Free Lie Algebras[END_REF], L satisfies Friedrichs criterion [START_REF] Hoang Ngoc Minh | Polylogarithms and Shuffle Algebra[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF] :

∀u, v ∈ X * , Li u⊔⊔ v = Li u Li v , ( 36
) ⇒ ∀u, v ∈ x 0 X * x 1 , ζ(u⊔⊔ v) = ζ(u) ζ(v). ( 37
)
Proposition 3 (Successive differentiation of L, [START_REF] Ngoc | Differential Galois groups and noncommutative generating series of polylogarithms[END_REF]). For any l ∈ N, let

P l (z) = wgt(r)=l w∈X deg(r) deg(r) i=1 i j=1 r i + j -1 r i τ r (w) ∈ C X ,
where, for any w = x i1 • • • x i k and r = (r 1 , . . . , r k ) of degree deg(r) = k and of weight wgt(r

) = k + r 1 + • • • + r k , the polynomial τ r (w) = τ r1 (x i1 ) • • • τ r k (x i k ) is defined by ∀r ∈ N, τ r (x 0 ) = ∂ r x 0 z = -r!x 0 (-z) r+1 and τ r (x 1 ) = ∂ r x 1 1 -z = r!x 1 (1 -z) r+1 . Denoting ∂ = d/dz, we have ∂ l L(z) = P l (z)L(z).
Let { Šl } l∈LynX be the transcendence basis of the shuffle algebra (Q X , ⊔⊔ ) and { Šw } w∈X * be the associated completed basis of the shuffle algebra (Q X , ⊔⊔ ) (viewed as a Q-module). They are defined as follows [START_REF] Reutenauer | Free Lie Algebras[END_REF]

Š1 X * = 1 for l = 1 X * (38) Šl = x Šu , for l = xu ∈ LynX, (39) Šw
= Š⊔⊔ i1 l1 ⊔⊔ . . . ⊔⊔ Š⊔⊔ i k l k i 1 ! . . . i k ! for w = l i1 1 . . . l i k k , l 1 > . . . > l k . (40)
Let {S w } w∈Y * be the PBW basis of the envelopping algebra U(Lie Q X ) in duality with the basis { Šw } w∈Y * and {S l } l∈LynX is then the basis of the Lie algebra Lie Q X [START_REF] Reutenauer | Free Lie Algebras[END_REF].

Theorem 3 (Factorization of L, [START_REF] Hoang Ngoc Minh | Polylogarithms and Shuffle Algebra[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF]). Let

L reg := ց l∈LynX-X e LiS l Šl .
Then L(z) = e -x1 log(1-z) L reg (z)e x0 log z .

For l ∈ LynX -X, the polynomial S l is a finite combination of words in x 0 X * x 1 . Then we can state the following Definition 2 ( [START_REF] Hoang Ngoc Minh | Polylogarithms and Shuffle Algebra[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF]). We set Z ⊔⊔ := L reg (1).

In the definitions 1 and 2 only convergent polyzêtas arise and these noncommutative generating series will induce, in Section 3.1, two algebra morphisms of regularization as shown in the theorems 8 and 9 respectively. Hence, these power series are quite different of those given in [START_REF] Lê | Kontsevich's integral for Kauffman polynomial[END_REF] or in [START_REF] Racinet | Doubles mélanges des polylogarithmes multiples aux racines de l'unité[END_REF] (the last is based on [4], see [START_REF] Cartier | Fonctions polylogarithmes, nombres polyzêtas et groupes prounipotents[END_REF]) needing a regularization process.

Asymptotic expansions by noncommutative generating series and regularized Chen generating series

Let ρ 1-z , ρ 1-1 z and ρ 1 z [START_REF] Hoang | L'algèbre des polylogarithmes par les séries génératrices[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF] be three monoid morphisms verifying

ρ 1-z (x 0 ) = -x 1 and ρ 1-z (x 1 ) = -x 0 , (41)
ρ 1-1/z (x 0 ) = -x 0 + x 1 and ρ 1-1/z (x 1 ) = -x 0 (42) ρ 1/z (x 0 ) = -x 0 + x 1 and ρ 1/z (x 1 ) = x 1 . (43)
Using homographic transformations belonging to the group G, one has [START_REF] Hoang | L'algèbre des polylogarithmes par les séries génératrices[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF] L(1

-z) = e x0 log(1-z) ρ 1-z [L reg (z)]e -x1 log z Z ⊔⊔ , (44)
L(1 -1/z) = e x0 log(1-z) ρ 1-1 z [L reg (z)]e -x1 log z ρ 1-1/z (Z -1 ⊔⊔ )e iπx0 (45)
L(1/z) = e -x1 log(1-z) ρ 1/z [L reg (z)]e (-x0+x1) log z ρ 1/z (Z -1 ⊔⊔ )e iπx1 Z ⊔⊔ . (46)
Thus, [START_REF] Hoang Ngoc Minh | Polylogarithms and Shuffle Algebra[END_REF] and ( 44) yield [START_REF] Hoang | L'algèbre des polylogarithmes par les séries génératrices[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF] L(z) z→0 exp(x 0 log z) and L(z) z→1 exp(-

x 1 log(1 -z)) Z ⊔⊔ . ( 47
)
Let us call LI C the smallest algebra containing C, closed under derivation and under integration with respect to ω 0 and ω 1 . It is the C-module generated by the polylogarithms {Li w } w∈X * .

Let π Y : LI C X -→ LI C Y be a projector such that for any f ∈ LI C and w ∈ X * , π Y (f wx 0 ) = 0. Then [START_REF] Ngoc | Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Inputs[END_REF]

Λ(z) = π Y L(z) z→1 exp y 1 log 1 1 -z π Y Z ⊔⊔ . (48)
Since the coefficient of z N in the ordinary Taylor expansion of P

y k 1 is H y k 1 (N ) then let Mono(z) := e -(x1+1) log(1-z) = k≥0 P y k 1 (z) y k 1 ( 49
) Const := k≥0 H y k 1 y k 1 = exp - k≥1 H y k (-y 1 ) k k . ( 50
)
Proposition 4 ( [START_REF] Ngoc | Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Inputs[END_REF]). We have

π Y P(z) z→1 Mono(z)π Y Z ⊔⊔ and H(N ) N →∞ Const(N )π Y Z ⊔⊔ .
Proof. Let µ be the morphism verifying µ(x 0 ) = x 1 and µ(x 1 ) = x 0 . Then, by Theorem 3, the noncommutative generating series of {P w } w∈X * is given by

P(z) = (1 -z) -1 L(z) = e -(x1+1) log(1-z) L reg (z)e x0 log z = e x0 log z µ[L reg (1 -z)]e -(x1+1) log(1-z) Z ⊔⊔ = e x0 log z µ[L reg (1 -z)]Mono(z)Z ⊔⊔ .
Thus, P(z) z→0 e x0 log z and P(z) z→1 Mono(z)Z ⊔⊔ yielding the expected results.

As consequence of ( 48)-( 50) and of Proposition 4, one gets Theorem 4 (à la Abel, [START_REF] Ngoc | Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Inputs[END_REF]).

lim z→1 exp y 1 log 1 1 -z Λ(z) = lim N →∞ exp k≥1 H y k (N ) (-y 1 ) k k H(N ) = π Y Z ⊔⊔ .
Therefore, the knowledge of the ordinary Taylor expansion at 0 of the polylogarithmic functions {P w (1z)} w∈X * gives Theorem 5 ([12]). For all g ∈ C{P w } w∈Y * , there exists algorithmically computable

c j ∈ C, α j ∈ Z, β j ∈ N and b i ∈ C, η i ∈ Z, κ i ∈ N such that g(z) z→1 +∞ j=0 c j (1 -z) αj log βj (1 -z) and [z n ]g(z) N →+∞ +∞ i=0 b i n ηi log κi (n).
Definition 3. Let Z be the Q-algebra generated by convergent polyzêtas and let Z ′ be the11 Q[γ]-algebra generated by Z.

Corollary 1 ([12]

). There exists algorithmically computable c j ∈ Z,

α j ∈ Z, β j ∈ N and b i ∈ Z ′ , κ i ∈ N, η i ∈ Z such that ∀w ∈ Y * , P w (z) ∼ +∞ j=0 c j (1 -z) αj log βj (1 -z) for z → 1, ∀w ∈ Y * , H w (N ) ∼ +∞ i=0 b i N ηi log κi (N )
for N → +∞.

The Chen generating series along the path z 0 z, associated to ω 0 , ω 1 is the following

S z0 z := w∈X * S | w w with S | w = α z z0 (w) (51)
which solves the differential equation [START_REF] Ngoc | Lyndon words, polylogarithmic functions and the Riemann ζ function[END_REF] with the initial condition S z0 z0 = 1. Thus, S z0 z and L(z)L(z 0 ) -1 satisfy the same differential equation taking the same value at z 0 and

S z0 z = L(z)L(z 0 ) -1 . ( 52
)
Any Chen generating series S z0 z is group like [START_REF] Ree | Lie elements and an algebra associated with shuffles[END_REF] and depends only on the homotopy class of z 0 z [START_REF] Chen | Iterated path integrals[END_REF]. The product of S z1 z2 and S z0 z1 is the Chen generating series

S z0 z2 = S z1 z2 S z0 z1 . (53) ¡ ¡ s s s 0 ∞ 1 Figure 1: Hexagonal path Let ε ∈]0, 1[ and let z i = ε exp(iθ i ), for i = 0 or 1. We set θ = θ 1 -θ 0 . Let Γ 0 (ε, θ) (resp. Γ 1 (ε, θ
)) be the path turning around 0 (resp. 1) in the positive direction from z 0 to z 1 . By induction on the length of w, one has

| S Γi(ε,θ) | w | = (2ε) |w|x i θ |w|| /| w |!, (54)
where, | w | denotes the lenghth of w and | w | xi , denotes the number of occurrences of letter x i in w, for i = 0, 1. For ε → 0 + , these estimations yield

S Γi(ε,θ) = e iθxi + o(ε). (55)
In particular, if Γ 0 (ε) (resp. Γ 1 (ε)) is a circular path of radius ε turning around 0 (resp. 1) in the positive direction, starting at z = ε (resp. 1ε), then, by the noncommutative residu theorem [START_REF] Hoang Ngoc Minh | Polylogarithms and Shuffle Algebra[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF], we get

S Γ0(ε) = e 2iπx0 + o(ε) and S Γ1(ε) = e -2iπx1 + o(ε). (56)
Finally, the asymptotic behaviors of L on [START_REF] Schützenberger | On the definition of a family of automata[END_REF] give Proposition 5 ([32, 35]). We have S ε 1-ε ε→0 + e -x1 log ε Z ⊔⊔ e -x0 log ε .

In other terms, Z ⊔⊔ is the regularized Chen generating series S ε 1-ε of diffferential forms ω 0 and ω 1 : Z ⊔⊔ is the noncommutative generating series of the finite parts of the coefficients of the Chen generating series e x1 log ε S ε 1-ε e x0 log ε : the concatenation of e x0 log ε and then S ε 1-ε and finally, e x1 log ε . Proposition 6. Let ρ 1-1/z be the morphism is given in Section 2.2.2. We have

ց l∈LynX l =x 0 ,x 1 e ζ( ľ)l = e iπx0 ց l∈LynX l =x 0 ,x 1 e ζ( ľ)ρ 1-1/z (l) e iπ(-x0+x1) ց l∈LynX l =x 0 ,x 1 e ζ( ľ)ρ 2 1-1/z (l) e -iπx1 .
Proof. Following the hexagonal path given in Figure 1, one has [START_REF] Hoang | L'algèbre des polylogarithmes par les séries génératrices[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF] (

S ε 1-ε e iπx0 )ρ 1-1/z (S ε 1-ε e iπx0 )ρ 2 1-1/z (S ε 1-ε e iπx0 ) = 1 + O( √ ε).
By Proposition 5, it follows the hexagonal relation [START_REF] Drinfel | -Quasi-Hopf Algebras[END_REF][START_REF] Drinfel'd | -On quasitriangular quasi-hopf algebra and a group closely connected with gal(q/q)[END_REF][START_REF] Hoang | L'algèbre des polylogarithmes par les séries génératrices[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF] which is

Z ⊔⊔ e iπx0 ρ 1-1/z (Z ⊔⊔ )e iπ(-x0+x1) ρ 2 1-1/z (Z ⊔⊔ )e -iπx1 = 1, ⇐⇒ e iπx0 ρ 1-1/z (Z ⊔⊔ )e iπ(-x0+x1) ρ 2 1-1/z (Z ⊔⊔ )e -iπx1 = Z -1 ⊔⊔ .
It follows then the expected result. The left residual (resp. right residual) of S by P , is the formal power series P ⊳ S (resp. S ⊲ P ) in Q X defined by :

Indiscernability over a class of formal power series

P ⊳ S | w = S | wP (resp. S ⊲ P | w = S | P w ).
We straightforwardly get, for any P, Q ∈ Q X :

P ⊳ (Q ⊳ S) = P Q ⊳ S, (S ⊲ P ) ⊲ Q = S ⊲ P Q, (P ⊳ S) ⊲ Q = P ⊳ (S ⊲ Q). ( 57
)
In case x, y ∈ X and w ∈ X * , we get x ⊳ (wy) = δ x,y w and xw ⊲ y = δ x,y w.

Lemma 1 (Reconstruction lemma). Let S ∈ Q X . Then S = S | ǫ + x∈X x(S ⊲ x) = S | ǫ + x∈X (x ⊳ S)x.
Lemma 2. The left and right residuals by a letter x are derivations in (Q X , ⊔⊔ ) :

x ⊳ (u ⊔⊔ v) = (x ⊳ u) ⊔⊔ v + u ⊔⊔ (x ⊳ v), (u ⊔⊔ v) ⊲ x = (u ⊲ x) ⊔⊔ v + u ⊔⊔ (v ⊲ x).
Proof. Use the recursive definitions of the shuffle product.

Lemma 3. For any Lie polynomial Q ∈ Lie Q X , the linear maps "Q⊳" and "⊲Q" are derivations on (Q[LynX], ⊔⊔ ).

Proof. For any l, l 1 , l 2 ∈ LynX, we have

l ⊳ (l 1 ⊔⊔ l 2 ) = l 1 ⊔⊔ ( l ⊳ l 2 ) + ( l ⊳ l 1 ) ⊔⊔ l 2 = l 1 δ l2, l + δ l1, ll 2 , (l 1 ⊔⊔ l 2 ) ⊲ l = l 1 ⊔⊔ (l 2 ⊲ l) + (l 1 ⊲ l) ⊔⊔ l 2 = l 1 δ l2, l + δ l1, ll 2 .
Lemma 4. For any Lyndon word l ∈ LynX and Šl defined as in [START_REF] Ngoc | Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Inputs[END_REF], one has

x 1 ⊳ l = l ⊲ x 0 = 0 and x 1 ⊳ Šl = Šl ⊲ x 0 = 0.
Proof. Since x 1 ⊳ and ⊲x 0 are derivations and for any l ∈ LynX -X, the polynomial Šl belongs to x 0 Q X x 1 then it follows the expected results.

Theorem 6 (On representative series). The following properties are equivalent for any series S ∈ Q X :

1. The left C-module Res g (S) = span{w ⊳ S | w ∈ X * } is finite dimensional. 2. The right C-module Res d (S) = span{S⊲w | w ∈ X * } is finite dimensional. 3. There are matrices λ ∈ M 1,n (Q), η ∈ M n,1 (Q) and a representation of X * in M n,n , such that S = w∈X * [λµ(w)η] w = λ ց l∈LynX e µ(S l ) Šl η.
A series that safisfies the items of this theorem will be called representative series. This concept can be found in [START_REF] Abe | Hopf algebra[END_REF][START_REF] Hochschild | The structure of Lie groups[END_REF][START_REF] Duchamp | -Sweedler's duals and Schützenberger's calculus[END_REF]. The two first items are in [START_REF] Fliess | Matrices de Hankel[END_REF][START_REF] Hespel | Une étude des séries formelles noncommutatives pour l'Approximation et l'Identification des systèmes dynamiques[END_REF]. The third item can be deduced from [START_REF] Chari | A guide to quantum group[END_REF][START_REF] Duchamp | Un critère de rationalité provenant de la géométrie noncommutative[END_REF] for example and it was used to factorize first time, by Lyndon words, the output of bilinear and analytical dynamical systems respectively in [START_REF] Ngoc | Input/Output behaviour of nonlinear control systems : about exact and approximated computations[END_REF][START_REF] Ngoc | Input/Output Behaviour of Nonlinear Control Systems : Rational Approximations, Nilpotent structural Approximations[END_REF] and to study polylogarithms, hypergeometric functions and associated functions in [START_REF] Ngoc | Fonctions de Dirichlet d'ordre n et de paramètre t[END_REF][START_REF] Ngoc | Symbolic integration of meromorphic differential systems via Dirichlet functions[END_REF][START_REF] Ngoc | Differential Galois groups and noncommutative generating series of polylogarithms[END_REF]. The dimension of Res g (S) is equal to that of Res d (S), and to the minimal dimension of a representation satisfying the third point of Theorem 6. This rank is then equal to the rank of the Hankel matrix of S, that is the infinite matrix ( S | uv ) u,v∈X indexed by X * × X * and is also called Hankel rank of S [START_REF] Fliess | Matrices de Hankel[END_REF][START_REF] Hespel | Une étude des séries formelles noncommutatives pour l'Approximation et l'Identification des systèmes dynamiques[END_REF] : [START_REF] Fliess | Matrices de Hankel[END_REF][START_REF] Hespel | Une étude des séries formelles noncommutatives pour l'Approximation et l'Identification des systèmes dynamiques[END_REF]). The Hankel rank of a formal power series S ∈ C X is the dimension of the vector space

Definition 5 ([
{S ⊲ Π | Π ∈ C X }, (resp. {Π ⊳ S | Π ∈ C X }.
The triplet (λ, µ, η) is called a linear representation of S. We define the minimal representation 12 of S as being a representation of S of minimal dimension.

For any proper series S, the following power series is called "star of S"

S * = 1 + S + S 2 + . . . + S n + . . . . (58)
Definition 6 ([2, 47]). A series S is called rational if it belongs to the closure in Q X of the noncommutative polynomial algebra by sum, product and star operation of proper13 elements. The set of rational power series will be denoted by Q rat X .

Lemma 5. For any noncommutative rational series (resp. polynomial) R and for any polynomial P , the left and right residuals of R by P are rational (resp. polynomial).

Theorem 7 (Schützenberger, [START_REF] Berstel | Rational series and their languages[END_REF][START_REF] Schützenberger | On the definition of a family of automata[END_REF]). Any noncommutative power series is representative if and only if it is rational.

Continuity and indiscernability

Definition 7 ( [START_REF] Ngoc | Contribution au développement d'outils informatiques pour résoudre des problèmes d'automatique non linéaire[END_REF][START_REF] Ngoc | Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Inputs[END_REF]). Let H be a class of formal power series over X and let S ∈ C X .

1. S is said to be continuous14 over H if for any Φ ∈ H, the following sum, denoted S || Φ , is convergent in norm

w∈X * S | w Φ | w .
The set of continuous power series over H will be denoted by C cont X .

2. S is said to be indiscernable15 over H if and only if

∀Φ ∈ H, S || Φ = 0.
Let ρ be the monoid morphism verifying ρ(x 0 ) = x 1 and ρ(x 1 ) = x 0 and let ŵ = ρ( w), where w is the mirror of w.

Lemma 6. Let S ∈ C cont X . If S || Z ⊔⊔ = 0 then Ŝ || Z ⊔⊔ = 0, where Ŝ := w∈X * S | w ŵ.
Proof. For any w ∈ x 0 X * x 1 , by "duality relation", one has (see [START_REF] Hoffman | The Multiple harmonic series[END_REF][START_REF] Zagier | Values of zeta functions and their applications[END_REF][START_REF] Hoang | L'algèbre des polylogarithmes par les séries génératrices[END_REF])

ζ( ŵ) = ζ(w), or equivalently Z ⊔⊔ = Ẑ⊔⊔ := w∈X * Z ⊔⊔ | w ŵ. Using the fact Ŝ || Z ⊔⊔ = ŵ∈X * S | ŵ Z ⊔⊔ | ŵ = w∈X * S | w Z ⊔⊔ | w ,
one gets finally the expected result.

S || Φx = S || lim t→0 Φ e t x -1 t = lim t→0 S || Φ e t x -1 t , S || xΦ = S || lim t→0 e t x -1 t Φ = lim t→0 S || e t x -1 t Φ .
Since S is indiscernable over H then

S || Φx = lim t→0 1 t S || Φe t x -lim t→0 1 t S || Φ = 0, S || xΦ = lim t→0 1 t S || e t x Φ -lim t→0 1 t S || Φ = 0. Proposition 7.
Let H be a monoid containing {e t x } t∈C x∈X . The formal power series S ∈ C cont X is indiscernable over H if and only if S = 0.

Proof. If S = 0 then it is immediate that S is indiscernable over H. Conversely, if S is indiscernable over H then by Lemma 7, for any word w ∈ X * , by induction on the length of w, w ⊳ S is indiscernable over H and then in particular,

w ⊳ S || Id H = S | w = 0.
In other words, S = 0.

3 Group of associators : polynomial relations among convergent polyzêtas and identification of local coordinates ) → (R, .) be the morphism verifying the following properties

• for u, v ∈ Y * , ζ (u v) = ζ (u)ζ (v), • for all convergent word w ∈ Y * -y 1 Y * , ζ (w) = ζ(w), • ζ (y 1 ) = 0. Then w∈X * ζ (w) w = Z . Corollary 2 ([33]). For any w ∈ X * , ζ (w) belongs to the algebra Z. Theorem 9 ([33]). Let ζ ⊔⊔ : (Q X , ⊔⊔ ) → (R, .
) be the morphism verifying the following properties

• for u, v ∈ X * , ζ ⊔⊔ (u⊔⊔ v) = ζ ⊔⊔ (u)ζ ⊔⊔ (v), • for all convergent word w ∈ x 0 X * x 1 , ζ ⊔⊔ (w) = ζ(w), • ζ ⊔⊔ (x 0 ) = ζ ⊔⊔ (x 1 ) = 0. Then w∈X * ζ ⊔⊔ (w) w = Z ⊔⊔ .

Corollary 3 ([33]

). For any w ∈ Y * , ζ ⊔⊔ (w) belongs to the algebra Z. Definition 8. For any w ∈ Y * , let γ w be the constant part 16 of the asymptotic expasion, on the comparison scale {n a log b (n)} a∈Z,b∈N , of H w (n).

Let Z γ be the noncommutative generating series of {γ w } w∈Y * :

Z γ := w∈Y * γ w w.
Definition 9. We set

B(y 1 ) := exp - k≥1 γ y k (-y 1 ) k k and B ′ (y 1 ) := e -γy1 B(y 1 ).
The power series B ′ (y 1 ) corresponds in fact to the mould 17 Mono in [START_REF] Gari | la dimorphie et l'arithmétique des multizêtas : un premier bilan[END_REF] and to the Φ corr in [START_REF] Racinet | Doubles mélanges des polylogarithmes multiples aux racines de l'unité[END_REF] (see also [4,[START_REF] Cartier | Fonctions polylogarithmes, nombres polyzêtas et groupes prounipotents[END_REF]). While the power series B(y 1 ) corresponds to the Gamma Euler function with its product expansion,

B(y 1 ) = Γ(y 1 + 1), 1 Γ(y 1 + 1) = e γy1 n≥1 1 + y 1 n e -γ/n . ( 59
)
Lemma 8 ( [START_REF] Ngoc | Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Inputs[END_REF]). Let b n,k (t 1 , . . . , t n-k+1 ) be the (exponential) partial Bell polynomials in the variables {t l } l≥1 given by the exponential generating series

exp u ∞ l=0 t l v l l! = ∞ n,k=0 b n,k (t 1 , . . . , t n-k+1 ) v n u k n! .
For any m ≥ 1, let t m = (-1) m (m -1)!γ ym . Then

B(y 1 ) = 1 + n≥1 n k=1 b n,k (γ, -ζ(2), 2ζ(3), . . .) (-y 1 ) n n! .
Since the ordinary generating series of the finite parts of coefficients of Const(N ) is nothing else but the power series B(y 1 ), taking the constant part on either side of H(N ) N →∞ Const(N )π Y Z ⊔⊔ (see Proposition 4), yields Theorem 10 ( [START_REF] Ngoc | Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Inputs[END_REF]). We have Z γ = B(y 1 )π Y Z ⊔⊔ . 16 i.e. γw is the Euler-Mac Laurin constante of Hw(n). 17 The readers can see why we have introduced the power series Mono(z) in Proposition 4.

Identifying the coefficients of y k

1 w on either side using the identity 18 [START_REF] Hoang | De l'algèbre des ζ de Riemann multivariées l'algèbre des ζ de Hurwitz multivariées[END_REF] ∀u

∈ X * x 1 , x k 1 x 0 u = k l=0 x l 1 ⊔⊔ (x 0 [(-x 1 ) k-l ⊔⊔ u]) (60)
and applying the morphism ζ ⊔⊔ given in Theorem 9, we get [START_REF] Hoang | De l'algèbre des ζ de Riemann multivariées l'algèbre des ζ de Hurwitz multivariées[END_REF] ∀u

∈ X * x 1 , ζ ⊔⊔ (x k 1 x 0 u) = ζ(x 0 [(-x 1 ) k ⊔⊔ u]). ( 61
) Corollary 4 ([39]). For w ∈ x 0 X * x 1 , i.e. w = x 0 u and π Y w ∈ Y * -y 1 Y * , and for k ≥ 0, the constant γ (x k 1 w) associated to the divergent polyzêta ζ(x k 1 w
) is a polynomial of degree k in γ and with coefficients in Z :

γ x k 1 w = k i=0 ζ(x 0 [(-x 1 ) k-i ⊔⊔ u]) i! i j=1 b i,j (γ, -ζ(2), 2ζ(3), . . .) .
Moreover, for l = 0, .., k, the coefficient of γ l is of weight | w | +kl.

In particular, for s > 1, the constant γ y1ys associated to ζ(y 1 y s ) is linear in γ and with coefficients in

Q[ζ(2), ζ(2i + 1)] 0<i≤(s-1)/2 . Corollary 5 ([39]). The constant γ x k 1 associated to the divergent polyzêta ζ(x k 1 ) is a polynomial of degree k in γ with coefficients in Q[ζ(2), ζ(2i+1)] 0<i≤(k-1)/2 : γ x k 1 = s 1 ,...,s k ≥0 s 1 +...+ks k =k+1 (-1) k s 1 ! . . . s k ! (-γ) s1 - ζ(2) 2 s2 . . . - ζ(k) k s k .
Moreover, for l = 0, .., k, the coefficient of γ l is of weight kl.

We thereby obtain the following algebra morphism, denoted by γ • , for the regularization to γ with respect to the quasi-shuffle product independently to the regularization with respect to the shuffle product 19 and then by applying the tensor product of morphisms γ • ⊗ Id on the diagonal series, over Y , we get (see Annexe A) Theorem 11. The mapping γ • realizes the morphism from (Q Y , ) to (R, .) verifying the following properties 18 By the Convolution Theorem [START_REF] Ngoc | Summations of Polylogarithms via Evaluation Transform[END_REF], this is equivalent to

∀u ∈ X * , α z 0 (x k 1 x 0 u) = z 0 [log(1 -s) -log(1 -z)] k k! α s 0 (u) ds s = k l=0 [-log(1 -z)] l l! z 0 log k-l (1 -s) (k -l)! α s 0 (u) ds s .
This theorem induces de facto the algebra morphism of regularization to 0 with respect to the shuffle product, as shown the Theorem 9. 19 In [4,[START_REF] Cartier | Fonctions polylogarithmes, nombres polyzêtas et groupes prounipotents[END_REF][START_REF] Ihara | Derivation and double shuffle relations for multiple zetas values[END_REF][START_REF] Waldschmidt | Hopf Algebra and Transcendental numbers[END_REF], the authors suggest the simultaneous regularizations, with respect to the shuffle product and the quasi-shuffle product, to T and then to set T = 0.

• for any word

u, v ∈ Y * , γ u v = γ u γ v , • for any convergent word w ∈ Y * -y 1 Y * , γ w = ζ(w) • γ y1 = γ.
Then Z γ = e γy1 Z .

Identities of noncommutative generating series of polyzêtas

Corollary 6. With the notations of Definition 9, we have

Z γ = B(y 1 )π Y Z ⊔⊔ ⇐⇒ Z = B ′ (y 1 )π Y Z ⊔⊔ , π Y Z ⊔⊔ = B -1 (x 1 )Z γ ⇐⇒ Z ⊔⊔ = B ′-1 (x 1 )π X Z .
Roughly speaking, for the quasi-shuffle product, the regularization to γ is "equivalent" to the regularization to 0.

Note also that the constant γ y1 = γ is obtained as the finite part of the asymptotic expansion of H 1 (n) in the comparison scale {n a log b (n)} a∈Z,b∈N .

In the same way, since n and H 1 (n) are algebraically independent, as arithmetical functions (see Proposition 1), then {n a H b 1 (n)} a∈Z,b∈N constitutes a new comparison scale for asymptotic expansions.

Hence, the constants ζ ⊔⊔ (x 1 ) = 0 and ζ (y 1 ) = 0 can be interpreted as the finite part of the asymptotic expansions of Li 1 (z) and H 1 (n) respectively in the comparison scales

{(1 -z) a log(1 -z) b } a∈Z,b∈N and {n a H b 1 (n)} a∈Z,b∈N . Definition 10 ([33]). Let C 1 := Qǫ ⊕ x 0 Q X x 1 , C 2 := Qǫ ⊕ (Y -{y 1 })Q Y .
Lemma 9 ( [START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF][START_REF] Hoang | De l'algèbre des ζ de Riemann multivariées l'algèbre des ζ de Hurwitz multivariées[END_REF]). We get

(C 1 , ⊔⊔ ) ∼ = (C 2 , ).
Using a theorem of Radford [START_REF] Reutenauer | Free Lie Algebras[END_REF] and its analogous over Y (see Annexe A), we get Proposition 8 ( [START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF][START_REF] Hoang | De l'algèbre des ζ de Riemann multivariées l'algèbre des ζ de Hurwitz multivariées[END_REF]).

(Q X , ⊔⊔ ) ∼ = (Q[LynX], ⊔⊔ ) = C 1 [x 0 , x 1 ], (Q Y , ) ∼ = (Q[LynY ], ) = C 2 [y 1 ],
This insures the effective way to get the finite part of the asymptotic expansions, in the comparison scales {(1-z) a log(1-z) b } a∈Z,b∈N and {n a H b 1 (n)} a∈Z,b∈N , of {Li w (z)} w∈Y * and {H w (N )} w∈Y * respectively. Proposition 9 ( [START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF][START_REF] Hoang | De l'algèbre des ζ de Riemann multivariées l'algèbre des ζ de Hurwitz multivariées[END_REF]). The restrictions of ζ ⊔⊔ and ζ over (C 1 , ⊔⊔ ) and (C 2 , ) respectively coincide with the following surjective algebra morphism

ζ : (C 2 , ) (C 1 , ⊔⊔ ) -→ (R, .) y r1 . . . y r k x 0 x r1-1 1 . . . x 0 x r k -1 1 -→ n1>...>n k >0 1 n r1 1 . . . n r k k ,
In Section 3.3 we will give the complete description of the kernel ker ζ.

With the double regularization 20 to zero [4,[START_REF] Cartier | Fonctions polylogarithmes, nombres polyzêtas et groupes prounipotents[END_REF][START_REF] Hoang | De l'algèbre des ζ de Riemann multivariées l'algèbre des ζ de Hurwitz multivariées[END_REF][START_REF] Racinet | Doubles mélanges des polylogarithmes multiples aux racines de l'unité[END_REF], the Drindfel'd associator Φ KZ corresponds then to Z ⊔⊔ (obtained with only convergent polyzêtas) as being the unique group-like element satisfying [START_REF] Hoang Ngoc Minh | Polylogarithms and Shuffle Algebra[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF]

Z ⊔⊔ | x 0 = Z ⊔⊔ | x 1 = 0 and ∀x ∈ x 0 X * x 1 , Z ⊔⊔ | w = ζ(w). ( 62
)
As consequence of Proposition 2, one has Proposition 10 ( [START_REF] Ngoc | Differential Galois groups and noncommutative generating series of polylogarithms[END_REF]).

log

Z ⊔⊔ = w∈X * ζ ⊔⊔ (w) π 1 (w), = k≥1 (-1) k-1 k u1,...,u k ∈X * -{ǫ} ζ ⊔⊔ (u 1 ⊔⊔ . . . ⊔⊔ u k ) u 1 . . . u k .
The associator Φ KZ can be also graded in the adjoint basis of U(Lie Q X ) as follows Proposition 11 ([38]). For any l ∈ N and P ∈ C X , let • denotes the composite operation defined by x 1 x l 0 • P = x 1 (x l 0 ⊔⊔ P ). Then

Z ⊔⊔ = k≥0 l1,••• ,l k ≥0 ζ ⊔⊔ (x 1 x l1 0 • • • • • x 1 x l k 0 ) k i=0 ad li x0 x 1 ,
where

ad l x0 x 1 is iterated Lie bracket ad l x0 x 1 = [x 0 , ad l-1 x0 x 1 ] and ad 0 x0 x 1 = x 1 .
Using the following expansion [6] ad

n x0 x 1 = n i=0 i n x n-i 0 x 1 x i 0 , (63)
one deduces then, via the regularization process of Theorem 9, the expression of the Drindfel'd associator Φ KZ given by Lê and Murakami [START_REF] Lê | Kontsevich's integral for Kauffman polynomial[END_REF].

Action of differential Galois group of polylogarithms on their asymptotic expansions

Group of associators theorem

Let A a be a commutative Q-algebra.

Since the polyzêtas satisfy [START_REF] Hoang | L'algèbre des polylogarithmes par les séries génératrices[END_REF], then by the Friedrichs criterion we can state the following Definition 11. Let dm(A) be the set of Φ ∈ A X such that 21

Φ | ǫ = 1, Φ | x 0 = Φ | x 1 = 0, ∆ ⊔⊔ Φ = Φ ⊗ Φ and such that, for Ψ = B ′ (y 1 )π Y Φ ∈ A Y then 22 ∆ Ψ = Ψ ⊗ Ψ.
Proposition 12 ([38]). If G(z) and H(z) are exponential solutions of (DE) then there exsists a Lie series

C ∈ Lie C X such that G(z) = H(z) exp(C).
Proof. Since H(z)H(z) -1 = 1 then by differentiating, we have

d[H(z)]H(z) -1 = -H(z)d[H(z) -1 ]. Therefore if H(z) is solution of Drinfel'd equation then d[H(z) -1 ] = -H(z) -1 [dH(z)]H(z) -1 = -H(z) -1 [x 0 ω 0 (z) + x 1 ω 1 (z)], d[H(z) -1 G(z)] = H(z) -1 [dG(z)] + [dH(z) -1 ]G(z) = H(z) -1 [x 0 ω 0 (z) + x 1 ω 1 (z)]G(z) -H(z) -1 [x 0 ω 0 (z) + x 1 ω 1 (z)]G(z).
By simplification, we deduce then H(z) -1 G(z) is a constant formal power series.

Since the inverse and the product of group like elements is group like then we get the expected result.

The differential C-module C{Li w } w∈X * is the universal Picard-Vessiot extension of every linear differential equations, with coefficients in C and admitting {0, 1, ∞} as regular singularities. The universal differential Galois group, noted by Gal(LI C ), is the set of differential C-automorphisms of C{Li w } w∈X * (i.e the automorphisms of C{Li w } w∈X * that let C point-wise fixed and that commute with derivation). The action of an automorphism of Gal(LI C ) can be determined by its action on Li w , for w ∈ X * . It can be resumed as its action on the noncommutative generating series L [START_REF] Ngoc | Differential Galois groups and noncommutative generating series of polylogarithms[END_REF] :

Let σ ∈ Gal(LI C ). Then

= x i l i1 1 • • • l i k k with l 1 > • • • > l k , one has σ Li Šl = ω xi σ Li i1 Šl 1 i 1 ! • • • σ Li i k Šl k i k ! + c Šl , ( 65
)
21 ∆ ⊔⊔ denotes the co-product of the shuflle product. 22 ∆ denotes the co-product of the quasi-shuflle product.

where c Šl is a constant of integration. For example,

σ Li x0x1 = Li x0x1 +c x1 Li x0 +c x0x1 , (66)
σ Li x 2 0 x1 = Li x 2 0 x1 + c x1 2 Li 2 x0 +c x0x1 Li x0 +c x 2 0 x1 , (67)
σ Li x0x 2 1 = Li x0x 2 1 +c x1 Li x0x1 + c 2 x1 2 Li x0 +c x0x 2 1 . (68)
Consequently,

w∈X *
σ Li w w = Le Cσ where e Cσ :=

ց l∈LynX e c Šl S l . ( 69
)
The action of σ ∈ Gal(LI C ) over {Li w } w∈X * is then equivalent to the action of the Lie exponential e Cσ ∈ Gal(DE) over the exponential solution L. So,

Theorem 12 ([38]). We have Gal(LI C ) = {e C | C ∈ Lie C X }.
Typically, since L(z 0 ) -1 is group-like then S z0 z = L(z)L(z 0 ) -1 is an other solution of (34) as already saw in (52).

Theorem 13 (Group of associators theorem). Let Φ ∈ A X and Ψ ∈ A Y be group-like elements, for the co-products ∆ ⊔⊔ , ∆ respectively, such that Ψ = B(y 1 )π Y Φ. There exists an unique

C ∈ Lie A X such that Φ = Z ⊔⊔ e C and Ψ = B(y 1 )π Y (Z ⊔⊔ e C ).
Proof. If C ∈ Lie A X then L ′ = Le C is group-like, for the co-product ∆ ⊔⊔ , and e C ∈ Gal(DE). Let H ′ be the noncommutative generating series of the Taylor coefficients, belonging to the harmonic algebra, of {(1z) -1 L ′ | w } w∈Y * . Then H ′ (N ) is also group-like, for the co-product ∆ . By the asymptotic expansion of L, we have L ′ (z) ε→1 e -x1 log(1-z) Z ⊔⊔ e C [START_REF] Hoang | L'algèbre des polylogarithmes par les séries génératrices[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF]. We put then Φ := Z ⊔⊔ e C and we deduce that

L ′ (z) 1 -z z→1 Mono(z)Φ and H ′ (N ) N →∞ Const(N )π Y Φ,
where the expressions of Mono(z) and Const(N ) are given on ( 49) and ( 50) respectively. Let κ w be the constant part of H ′ w (N ). Then

w∈Y * κ w w = B(y 1 )π Y Φ.
We put then Ψ := B(y 1 )π Y Φ (and also Ψ ′ := B ′ (y 1 )π Y Φ).

Corollary 7. We have

dm(A) = {Z ⊔⊔ e C | C ∈ Lie A X and e C | ǫ = 1, e C | x 0 = e C | x 1 = 0}.
Proof. On the one hand, Φ Therefore, Theorem 14. While Φ describes dm(A), the identities Ψ = B(y 1 )π Y Φ describe the ideal of polynomial relations, of coefficients in A, among generators of the A-algebra of convergent polyzêtas. Moreover, if the Euler constant, γ, does not belong to A then these relations are algebraically independent on γ.

| x 0 = Z ⊔⊔ | x 0 = 0, Φ | x 1 = Z ⊔⊔ | x 1 =
Simplyfied computations on Section 3.3 is an example of such identities. Some consequences of Theorem 14 will be drawn in Section 4.2.

Concatenation of Chen generating series

As an example of the action of the differential Galois group of polylogarithms on their asymptotic expansions, we are interrested on the action of their monodromy group which is contained in Gal(DE).

The monodromies at 0 and 1 of L are given respectively by [START_REF] Hoang Ngoc Minh | Polylogarithms and Shuffle Algebra[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF] M 0 L = Le 2iπm0 and

M 1 L = LZ -1 ⊔⊔ e -2iπx1 Z ⊔⊔ = Le 2iπm1 , (70)
where

m 0 = x 0 and m 1 = ց l∈LynX-X e -ζ( Šl ) adS l (-x 1 ). ( 71
) • If C = 2iπm 0 then Φ = Z ⊔⊔ e 2iπx0 , (72)
Ψ = exp γy 1 - k≥2 ζ(k) (-y 1 ) k k π Y Z ⊔⊔ (73) = Z . ( 74
)
The monodromy at 0 consists in the multiplication on the right of Z ⊔⊔ by e 2iπx0 and does not modify Z .

• If C = 2iπm 1 then Φ = e -2iπx1 Z ⊔⊔ , (75)
Ψ = exp (γ -2iπ

T := )y 1 - k≥2 ζ(k) (-y 1 ) k k π Y Z ⊔⊔ (76) = e -2iπy1 Z . ( 77
)
The monodromy at 1 consists in the multiplication on left of Z ⊔⊔ and of Z by e -2iπx1 and e -2iπy1 respectively.

Remark 1.

1. The monodromies around singularities of L could not allow, in this case, neither to introduce the factor e γx1 on the left of Z ⊔⊔ nor to eliminate the left factor e γy1 in Z γ (by putting 23 T = 0, for example).

2. By Proposition 5, we already saw that Z ⊔⊔ is the concatenation of Chen generating series [10] e x0 log ε and then S ε 1-ε and finally, e x1 log ε :

Z ⊔⊔ ε→0 + e x1 log ε S ε 1-ε e x0 log ε . ( 78
)
From ( 72) and (75), the action of the monodromy group gives

e x1 2k1iπ Z ⊔⊔ e x0 2k0iπ ε→0 + e x1(log ε+2k1iπ) S ε 1-ε e x0(log ε+2k0iπ) , ( 79
)
23 Why ?

as being the concatenation of the Chen generating series e x0(log ε+2k0iπ) (along circular path turning k 0 times around 0), then the Chen generating series S ε 1-ε and finally, the Chen generating series e x1(log ε+2k1iπ) (along circular path turning k 1 times around 1).

3. More generally, by Corollary 7, the action of the Galois differential group of polylogarithms states, for any Lie series C, the associator Φ = Z ⊔⊔ e C is the concatenation of some Chen generating series e C and e x0 log ε and then the Chen generating series S ε 1-ε and finally, e x1 log ε :

Φ ε→0 + e x1 log ε S ε 1-ε e x0 log ε e C . ( 80
)
By construction (see Theorem 13) the associator Φ is then the noncommutative generating series of the finite parts of the coefficients of the Chen generating series S z0 1-z0 e C , for z 0 = ε → 0 + . Hence, Corollary 9. Let Φ ∈ dm(A). For any differential produced formal power series S over X, there exists24 a differential representation (A, f ) such that :

Φ || S = w∈X * Φ | w A(w) • f |0 = ց l∈LynX-X e Φ| Šl A(S l ) • f |0 .

Algebraic combinatorial studies of polynomial relation among polyzêta via a group of associators

Here, Ȳ = {y 1 } ∪ {ȳ k } k≥2 . With the factorization of the monoids X * and Ȳ * by Lyndon words, let { l} l∈LynX and { l} l∈Lyn Ȳ be the dual of the Lyndon basis over X and Ȳ .

Preliminary study

As in Definition 10, let

A 1 = Aǫ ⊕ x 0 A X x 1 and A 2 = Aǫ ⊕ ( Ȳ -{y 1 })A Ȳ . (81)
For Φ ∈ dm(A), let Ψ = B ′ (y 1 )π Ȳ Φ. Let us introduce two algebra morphisms

φ : (A 1 , ⊔⊔ ) -→ A, u -→ Φ | u , ψ : (A 2 , ) -→ A, v -→ Ψ | v , (82)
verifying respectively φ(ǫ) = 1, φ(x 0 ) = φ(x 1 ) = 0 and ψ(ǫ) = 1, ψ(y 1 ) = 0.

Lemma 12. For any Φ ∈ dm(A), let Ψ = B ′ (y 1 )π Y Φ. Then ∀w ∈ Ȳ * -y 1 Ȳ * , ψ(w) = φ(π X w),
or equivalently, ∀w ∈ x 0 X * x 1 , φ(w) = ψ(π Ȳ w).

Lemma 13. We have

Φ = u∈X * φ(u) u = ց l∈LynX-X e φ(l) l and Ψ = v∈ Ȳ * ψ(u) u = ց l∈Lyn Ȳ -{y1} e ψ(l) l.
With the notations in Lemma 13, we can state the following Definition 12. We put

R := Φ∈dm(A)
ker φ (resp.

Ψ=B ′ (y 1 )π Ȳ Φ Φ∈dm(A) ker ψ).

Lemma 14. For any

Φ ∈ dm(A), let Ψ = B ′ (y 1 )π Ȳ Φ. Let Q ∈ Q[LynX] (resp. Q[Lyn Ȳ ]). Then Q || Φ = 0 ⇐⇒ Q ∈ ker φ (resp. Q || Ψ = 0 ⇐⇒ Q ∈ ker ψ).
Or equivalently (see Definition 7),

Q ∈ R ⇐⇒ Q is indiscernable over dm(A).
Let Φ 1 , Φ 2 ∈ dm(A). By Corollary 7, for i = 1 or 2, there exists an unique P i ∈ Lie A X such that e -Pi is well defined and

Φ i = Z ⊔⊔ e Pi , or equivalently, Z ⊔⊔ = Φ 1 e -P1 = Φ 2 e -P2 . (83)
Then, we get Φ 1 = Φ 2 e P1-P2 and Φ 2 = Φ 1 e P2-P1 . By Lemma 10, it follows Lemma 15. Let Φ 1 and Φ 2 ∈ dm(A). For any convergent Lyndon word, l, there exists a finite set I l ⊂ {λ ∈ LynX -X s.t. |λ| ≤ |l|} and the coefficients {p ′ i,u } u∈I l and {p ′′ i,u } u∈I l , for i = 1 or 2, belonging to A such that

φ i (l) = u∈I l p ′ i,u ζ(u), or equivalently, ζ(l) = u∈I l p ′′ i,u φ i (u).
There also exists the coefficients {p ′ u } u∈I l and {p ′′ u } u∈I l belonging to A such that

φ 1 (l) = u∈I l p ′ u φ 2 (u), or equivalently, φ 2 (l) = u∈I l p ′′ u φ 1 (u).
Therefore, the {φ i (l)} l∈LynX-X (resp. {ψ i (l)} l∈Lyn Ȳ -{y1} ), for i = 1 or 2, are also generators of the A-algebra generated by convergent polyzêtas.

Description of polynomial relations among coefficients of associator and irreducible polyzêtas

Since the identities of Corollary 8 (see also Corollary 6) hold for any pair of bases, in duality, compatible with factorization of the monoid X * (resp. Ȳ * ) then, by Corollary 8, one gets Theorem 15. For any Φ ∈ dm(A), let Ψ = B ′ (y 1 )π Ȳ Φ. We have

ց l∈Lyn Ȳ -y1 e ψ(l) l = exp k≥2 ζ(k) (-y 1 ) k k π Ȳ ց l∈LynX-X e φ(l) l.
If Φ = Z ⊔⊔ and Ψ = Z then, for ℓ ∈ LynX -X (resp. Lyn Ȳy 1 ), one has ζ(l) = φ(l) (resp. ψ(l)). Hence, one ontains (see also Corollary 6) Theorem 16 (Bis repetita).

ց l∈Lyn Ȳ -y1 e ζ(l) l = exp k≥2 ζ(k) (-y 1 ) k k π Ȳ ց l∈LynX-X e ζ(l) l.
Corollary 10. For any ℓ ∈ Lyn Ȳy 1 (resp. LynX -X), let

P ℓ ∈ U(Lie Q X ) (resp. U(Lie Q Ȳ )) be the decomposition of the polynomial π X l ∈ Q X (resp. π Ȳ l ∈ Q Ȳ ) in the PBW basis, induced by { l} l∈LynX (resp. { l} l∈Lyn Ȳ ), and let Pℓ ∈ Q[LynX -X] (resp. Q[Lyn Ȳ -y 1 ]
) be its dual. Then one obtains

π X ℓ -Pℓ ∈ ker φ (resp. π Ȳ ℓ -Pℓ ∈ ker ψ).
In particular, for φ = ζ (resp. ψ = ζ) then one also obtains

π X ℓ -Pℓ ∈ ker ζ (resp. π Ȳ ℓ -Pℓ ∈ ker ζ).
Moreover, for any ℓ ∈ Lyn Ȳ -y 1 (resp. LynX -X), the homogenous polynomial

π X ℓ -Pℓ ∈ Q X (resp. Q Ȳ ) is of degree equal | ℓ |≥ 2.
Proof. Since

ℓ ∈ Lyn Ȳ ⇐⇒ π X ℓ ∈ LynX -{x 0 }
then identifying the local coordinates (of second kind) on the two members of each identity in Theorem 15, one obtains

∀ℓ ∈ Lyn Ȳ -y 1 ⊂ Y * -y 1 Y * , ψ(ℓ) = φ( Pℓ ), (resp. ∀ℓ ∈ LynX -X ⊂ x 0 X * x 1 , φ(ℓ) = ψ( Pℓ )).
By Lemma 12, we get the expected result.

With the notations of Corollary 10, we get the following Definition 13. Let Q ℓ be the decomposition of the proper polynomial

π Ȳ ℓ -Pℓ (resp. π X ℓ -Pℓ ) in Lyn Ȳ (resp. LynX). Let R Ȳ := {Q ℓ } ℓ∈Lyn Ȳ -y1 and R X := {Q ℓ } ℓ∈LynX-X , L irr Ȳ := {ℓ ∈ Lyn Ȳ -y 1 | Q ℓ = 0} and L irr X := {ℓ ∈ LynX -X | Q ℓ = 0}.

It follows that

Lemma 16. We have

(Q[Lyn Ȳ -y 1 ], ) = (R Ȳ , ) ⊕ (Q[L irr Ȳ ], ), (Q[LynX -X], ⊔⊔ ) = (R X , ⊔⊔ ) ⊕ (Q[L irr X], ⊔⊔ ).
Then we can state the following Definition 14. Any word w is said to be irreducible if and only if w belongs to L irr Ȳ (resp. L irr X). In this case, the polyzêta ζ(w) est said to be Q-irreducible.

For any P ∈ Q[L irr X], there exists25 a differential representation (A, f ) such that P can be finitely factorized (see also Corollary 9) :

P = σf |0 = w∈X * irr A(w) • f w = ց ℓ∈Lirr X,finite e A( l) ℓ • f, (84)
where X * irr denotes the set of words obtaining by shuffling on L irr X. Lemma 17. Any proper polynomial

P ∈ (Q[L irr X], ⊔⊔ ) (resp. (Q[L irr Ȳ ], ))
is indiscernable over Chen generating series {e t x } t∈C x∈X : P || e t x0 = P || e t x1 = 0 (resp. P || e ty1 = 0).

Proof. By construction, x 0 and x 1 / ∈ L irr X (resp. y 1 / ∈ L irr X). For any n > 1, x n 0 and x n 1 (resp. y n 1 ) are not Lyndon words then they do not belong to L irr X (resp. L irr X). Therefore, for any n ≥ 0, one has P | x n 0 = P | x n 1 = 0 (resp. P | y n 1 = 0). Using the expansion of the exponential, we find the expected result.

Lemma 18. Let Φ ∈ dm(A) and let t ∈ C, x ∈ X. For any proper polynomial P ∈ (Q[L irr X], ⊔⊔ ), if P || Φ = 0 then P || Φe t x = 0 and P || e t x Φ = 0.

Proof. Since P ∈ (Q[L irr X], ⊔⊔ ) and P is proper then, by Lemma 17, for any t ∈ C and for any x ∈ X, we have P || e t x = 0 and then P || Φe t x = 0.

Since supp(P ) ⊂ x 0 X * x 1 then P || e t x0 Φ = P ⊲ e t x0 || Φ = 0. Next, for Φ ∈ dm(A), there exists e C such that e t x1 Φ = e t x1 Z ⊔⊔ e C and, by Proposition 5, we get

e t x1 Φ ε→0 + e x1(t+log ε) S ε 1-ε e x0 log ε e C .
Hence, there exists a Chen generating series C z 1-z0 and S z0 1-z0 such that we get the following asymptotic behaviour (see Section 3.2.2)

e t x1 Φ ε→0 + C z 1-z0 S z0 z e C
and the following concatenation holds [START_REF] Chen | Iterated path integrals[END_REF] (see Formula (53))

C z 1-z0 S z0 z = S z0 1-z0 , ⇐⇒ C z 1-z0 S z0 z e C = S z0 1-z0 e C .
Since P ∈ Q[L irr X] then by (84), applying σf |0 || • to the two sides of the previous equality, one has

σf |0 || C z 1-z0 S z0 z e C = σf |0 || S z0 1-z0 e C .
Thus, for z 0 = ε → 0 + , one obtains

σf |0 || e t x1 Φ ε→0 + σf |0 || Φ .
Since σf |0 || Φ = P || Φ = 0 then we get the expected result.

Lemma 19. For any Φ ∈ dm(A), let Ψ = B ′ (y 1 )π Ȳ Φ. We have R Ȳ ⊆ ker ψ and R X ⊆ ker φ. In particular, R Ȳ ⊆ ker ζ and R X ⊆ ker ζ.

Proposition 13. We have R X ⊆ R (resp. R Ȳ ⊆ R).

Proposition 14. For any proper polynomial

Q ∈ Q[L irr X] (resp. Q[L irr Ȳ ]), Q ∈ R ⇐⇒ Q = 0.
Proof. If Q = 0 then since, for Φ ∈ dm(A), φ is an algebra homorphism then

φ(Q) = 0. Hence, Q ∈ ker φ and then Q ∈ R. Conversely, if Q ∈ R then, for Φ ∈ dm(A), we get Q || Φ = 0. That means Q is indiscernable over dm(A).
Let H be the monoid generated by dm(A) and by the Chen generating series {e t x } t∈C x∈X . By Lemma 25, Q is continuous over H and by Lemma 18, it is indiscernable over H. By Proposition 7, the expected result follows.

Therefore, by the propositions 13 and 14, we obtain Theorem 17. We have R = R X (resp. R Ȳ ).

Proposition 15. For any

Φ ∈ dm(A), let Ψ = B ′ (y 1 )π Ȳ Φ. Let Q ∈ (Q[L irr X], ⊔⊔ ) (resp. (Q[L irr Ȳ ], )) such that Φ || Q = 0 (resp. Ψ || Q = 0). Then Q = 0.
Proof. Let H defined as being the monoid generated by Φ and by Chen generating series {e t x } t∈C x∈X . By assumption, Φ || Q = 0 and by Lemma 18, Q is then indiscernable over H. Finally, by Proposition 7, it follows that Q = 0.

Proposition 16. For any Φ ∈ dm(A), let Ψ = B ′ (y 1 )π Ȳ Φ. We get ker φ = R X (resp. ker ψ = R Ȳ ). In particular, ker ζ = R X (resp. ker ζ = R Ȳ ).
Proof. By Lemma 19, R X and R Ȳ are included in ker φ and ker ψ respectively.

Conversely, two cases can occur (see Lemma 16) :

1. Case Q / ∈ Q[L irr X] (resp. Q[L irr Ȳ ]). By Lemma 16, Q ≡ RX Q 1 (resp. Q ≡ R Ȳ Q 1 ) such that Q 1 ∈ Q[L irr X] (resp. Q[L irr Ȳ ]) and φ(Q 1 ) = 0 (resp. ψ(Q 1 ) = 0). This case is then reduced to the following 2. Case Q ∈ Q[L irr X] (resp. Q[L irr Ȳ ]). Using Proposition 15, we have Q ≡ RX 0 (resp. Q ≡ R Ȳ 0).
Then, R X (resp. R Ȳ ) contains ker φ (resp. ker ψ).

For any Q ∈ (Q[L irr X], ⊔⊔ ) (resp. (Q[L irr Ȳ ],
)), ζ(Q) is then a polynomial on A-irreducible polyzêtas (see Definition 14). Moreover, Proposition 17. The Q-algebra Z is generated by the family of A-irreducible polyzêtas {ζ(ℓ)} ℓ∈Lirr Ȳ (resp. {ζ(ℓ)} ℓ∈LirrX ).

Proof. By Radford's theorem [START_REF] Reutenauer | Free Lie Algebras[END_REF], one just needs to prove for Lyndon words :

Let ℓ ∈ Lyn Ȳ -y 1 . If π X ℓ = Pℓ then the result follows else one has

π X ℓ-Pℓ ∈ ker ζ. Hence, ζ(ℓ) = ζ( Pℓ ). Since Pℓ ∈ Q[LynX -X]
then Pℓ is polynomial on Lyndon words, over X, of degree less or equal | ℓ |. For each Lyndon word does appear in this decomposition of Pℓ , after applying π Ȳ , one uses the same recurcive procedure until getting Lyndon words in L irr Ȳ .

The same treatment works for any ℓ ′ ∈ LynX -X.

For any Φ ∈ dm(A), by Proposition 16, one also has ker φ = ker ζ = R X . That means, for any irreducible Lyndon words l = l ′ ,

φ(l) = φ(l ′ ) ⇐⇒ ζ(l) = ζ(l ′ ). ( 85
)
Let us state then the following Proof. The expected result follows by identifying coefficients in Φ = Z ⊔⊔ e C .

Finally, we can state the following Theorem 18. For any Φ ∈ dm(A), there exists an unique algebra homomorphism ϕ : Z -→ A such that Φ is computed from Z ⊔⊔ by applying ϕ to each coefficient :

Φ = w∈X * ϕ(ζ ⊔⊔ (w)) w = ց l∈LynX-X e ϕ(ζ(l)) l.
Remark 2.

1. In this work, neither the question deciding any real number belongs to Z or not nor the question expliciting {α l1,...,ln } n∈N l1,...,ln∈Lirr X in (86), are considered.

2. Now, by considering the commutative indeterminates t 1 , t 2 , t 3 , . . ., let A be the Q-algebra obtained by specializing Q[t 1 , t 2 , t 3 , . . .] at t 1 = iπ :

A = Q[iπ][t 2 , t 3 , . . .]. ( 87
)
Neither the Lie exponential series e iπx0 nor e iπx1 does belong to dm(A) but it belongs to Gal(DE). In particular, it figures in the modromies (see Section 3.2.2) or in the functional relations (see ( 45) and ( 46)) of polylogarithms and in the hexagonal relation of polyzêtas (see Proposition 6).

3. Applying Baker-Campbell-Hausdorff formula [6] to Proposition 6 we get, at orders 2 and 3 as examples, the famous Euler's formula saying ζ( 2) is an algebraic number over

A = Q[iπ] : ζ(2) + (iπ) 2 6 = 0 (order 2), (88)
ζ(3) -ζ(2, 1) = 0 (order 3, imaginary part). ( 89
)
Therefore, the first comming in mind homomorphism

ϕ : Z -→ A maps, at least ζ(2) to ϕ(ζ(2)) = π 2 /6.
4. For this reason, in [START_REF] Ngoc | Calcul symbolique non commutatif : aspects combinatoires des fonctions spéciales et des nombres spéciaux[END_REF], we have to consider the Q-algebra generated by iπ and by other A-irreducible polyzêtas obtained in [START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF][START_REF] Ngoc | Lyndon words, polylogarithmic functions and the Riemann ζ function[END_REF]3,[START_REF] Wardi | [END_REF] (and such algebra is denoted in this work by A). This algebra came up from the studies of monodromies [START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF][START_REF] Hoang Ngoc Minh | Polylogarithms and Shuffle Algebra[END_REF], as already shown in (70), and the Kummer type functional equations of polylogarithms [START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF][START_REF] Hoang | L'algèbre des polylogarithmes par les séries génératrices[END_REF], as already shown in ( 44)- [START_REF] Reutenauer | The local realisation of generating series of finite Lie rank[END_REF]. In particular, by ( 46), we get for example [START_REF] Hoang | L'algèbre des polylogarithmes par les séries génératrices[END_REF][START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF],

Li 2,1 1 t = - (iπ) 2 2 log t + iπ(ζ(2) - log 2 t 2 -Li 2 t) -Li 2,1 t + Li 3 t -log t Li 2 t + ζ(3) - log 3 t 6 . ( 90
)
specializing t = 1, the real part of this leads again to the Euler's indentity (89).

A = Q[iπ][t 2 , t 3 , . . .]. (91)
4.1 A conjecture by Pierre Cartier Definition 15 ([8,[START_REF] Racinet | Doubles mélanges des polylogarithmes multiples aux racines de l'unité[END_REF]). Let DM (A) denotes the set of Φ ∈ A X such that

Φ | ǫ = 1, Φ | x 0 = Φ | x 1 = 0, ∆ ⊔⊔ Φ = Φ ⊗ Φ
and such that, for

Ψ = exp - n≥2 π Y Φ | y n (-y 1 ) n n π Y Φ ∈ A Y , then ∆ Ψ = Ψ ⊗ Ψ.
Since DM (A) contains already Z ⊔⊔ then for Φ ∈ DM (A), by Theorem 13, there exists C ∈ Lie A X verifying

e C | ǫ = 1 and e C | x 0 = e C | x 1 = 0 such that Φ = Z ⊔⊔ e C ( 92
)
and such that

Ψ = B ′ (y 1 )π Y Φ = exp - k≥2 ζ(k) (-y 1 ) k k π Y Φ, (93)
Ψ = exp - n≥2 π Y Φ | y n (-y 1 ) n n π Y Φ. ( 94
)
By construction (see Definition 11 and Theorem 13), such Φ and Ψ are group-like (for the co-products ∆ ⊔⊔ and ∆ respectively) and here, Ψ must be also group-like (for the co-product ∆ ). If such a Lie series C exists then it is unique, due to the fact that e C = ΦZ -1 ⊔⊔ , and it is group-like (for the co-product ∆ ⊔⊔ ).

Corollary 11 (conjectured by Cartier, [START_REF] Cartier | Fonctions polylogarithmes, nombres polyzêtas et groupes prounipotents[END_REF]). For any Φ ∈ DM (A), there exists an unique algebra homomorphism 27 φ : Z -→ A such that Φ is computed from Z ⊔⊔ by applying φ to each coefficient.

Proof. By Theorem 18, use the fact

DM (Q) ⊆ DM (A) ⊆ dm(A).

Arithmetical nature of γ

By Theorem 14, under the assumption that the Euler constant, γ, does not belong to a commutative Q-algebra A then γ does not verify any polynomial with coefficients in A among the convergent polyzêtas. It follows then, 1. In the same spirit of Theorem 11, let ζ T be the regularization morphism 28 from (Q Y , ) to (R, .) mapping y 1 to T . Let Z T be the noncommutative generating series of polyzêtas regulariszed with respect to ζ T . Thus, as in Theorem 11 and by infinite factorization by Lyndon words, we also get

Z T := w∈X * ζ T (w) w = e T y1 Z . ( 95
)
2. Now let us consider the regularization, for N → +∞ and with respect to ζ T , of the power series Const(N ) given in ( 50) as

B T (y 1 ) = e T y1 B ′ (y 1 ) ( 96
)
27 See Remark 2(3) to have an example of φ. 28 This is a symbolic regularization and does not yet have an analytical justification as it is done, separately, for ζ ⊔⊔ and ζ in Section 3.1.2 as finite parts of the asymptotic expansions, in different scales of comparison, of Lix 1 (z), for z 1, and Hy 1 (N ), for N → ∞, respectively.

1. isomorphic to the graded algebra (A 1 /R X , ⊔⊔ ), or equivalently, (A 2 /R Ȳ , ).

2. freely generated by the A-irreducible polyzêtas {ζ(l)} l∈Lirr Ȳ (resp. {ζ(l)} l∈Lirr X ).

For any p ≥ 2, let

Z p = span Q {ζ(w) | w ∈ x 0 X * x 1 , | w |= p}. (108)
By definition of graded algebra [6], Theorem 19 means also that

Z = A ⊕ p≥2 Z p ( 109
)
and there is no linear relation among elements of different Z p ([8] conjecture C1, [START_REF] Waldschmidt | Hopf Algebra and Transcendental numbers[END_REF]). Thus, if θ is a (A-irreducible) polyzêta verifying the following algebraic equation Remark 4. In this work, neither the study of dim Z p [START_REF] Zagier | Values of zeta functions and their applications[END_REF] (see also [START_REF] Cartier | Fonctions polylogarithmes, nombres polyzêtas et groupes prounipotents[END_REF], conjecture C2) nor the estimate of the number of A-irreducible polyzêtas generating Z p , are discussed knowing the A-irreducible polyzêtas form transcendence basis of the A-algebra Z.

θ n + a n-1 θ n-1 + . . . + a 0 = 0 (110) then θ = 0 because Z p1 Z p2 ⊂ Z p1+p2 , for p 1 , p 2 ≥ 2,
5 Annexe A : pair of bases in duality and proof of Theorem 2

Preliminary results

Let Q Y be equipped the concatenation and the quasi-shuffle, , defined by

∀y i , y j ∈ Y = {y i } i≥1 , ∀u, v ∈ Y * , y i u y j v = y i (u y j v) + y i+j (y i u v), ∀w ∈ Y * , w 1 Y * = 1 Y * w = w,
or by its associated co-product, ∆ , defined by Proof. On one hand, since the P i 's are primitive then

∀y k ∈ Y, ∆ (y k ) = y k ⊗ 1 + 1 ⊗ y k + i+j=k y i ⊗ y j . satistying, for any u, v, w ∈ Y * , u ⊗ v | ∆ (w) = u v | w .
∆ (n-1) (P i ) = p+q=n-1 1 ⊗p ⊗ P i ⊗ 1 ⊗q .
On the other hand, S 1 . . .

S n | P 1 . . . P m = S 1 ⊗. . .⊗S n | ∆ (n-1) (P 1 . . . P m )
and ∆ (n-1) (P 1 . . . P m ) = ∆ (n-1) (P 1 ) . . . ∆ (n-1) (P m ). Hence,

S 1 . . . S n | P 1 . . . P m = n i=1 S i | m i=1 p+q=n-1 1 ⊗p ⊗ P i ⊗ 1 ⊗q .
1. For n > m, by expanding ∆ (n-1) (P 1 ) . . . ∆ (n-1) (P m ), one obtains a sum of tensors contening at least one factor equal to 1. For j = 1, .., n, the formal power series S j is proper and the result follows immediatly.

For

n = m, since n i=1 ∆ (n-1) (P i ) = σ∈Sn n i=1 P σ(i) + Q,
where Q is sum of tensors contening at least one factor equal to 1 and the S j ,'s are proper then S 1 ⊗ . . . ⊗ S n | Q = 0. Thus, the result follows.

3. For n < m, en tenant compte que, for j = 1, .., n, formal power series S j is proper, the expected follows by expanding the product

m i=1 ∆ (n-1) (P i ) = m i=1 p+q=n-1 1 ⊗p ⊗ P i ⊗ 1 ⊗q .
Proposition 18.

1. We have

log w∈Y * w ⊗ w = w∈Y + w ⊗ π 1 (w) = w∈Y + π * 1 (w) ⊗ w,
where π * 1 is the adjoint of π 1 and they are given by

π 1 (w) = k≥1 (-1) k-1 k u1,...,u k ∈Y + w | u 1 . . . u k u 1 . . . u k , π * 1 (w) = k≥1 (-1) k-1 k u1,...,u k ∈Y + w | u 1 . . . u k u 1 . . . u k .
In particular, for any y k ∈ Y , one has

π 1 (y k ) = y k + l≥2 (-1) l-1 l j 1 ,...,j l ≥1 j 1 +...+j l =k y j1 . . . y j l and π * 1 (y k ) = y k .
2. For any w ∈ Y * , we have

w = k≥0 1 k! u1,...,u k ∈Y * w | u 1 . . . u k π 1 (u 1 ) . . . π 1 (u k ), = k≥0 1 k! u1,...,u k ∈Y * w | u 1 . . . u k π * 1 (u 1 ) . . . π * 1 (u k ).
Proof.

1. Expanding the logarithm, we have log

w∈Y * w ⊗ w = k≥1 (-1) k-1 k w∈Y + w ⊗ w k = k≥1 (-1) k-1 k u1,...,u k ∈Y + (u 1 . . . u k ) ⊗ u 1 . . . u k = w∈Y + w ⊗ k≥1 (-1) k-1 k u1,...,u k ∈Y + w | u 1 . . . u k u 1 . . . u k .
In the same way, log

w∈Y * w ⊗ w = w∈Y + k≥1 (-1) k-1 k u1,...,u k ∈Y + w | u 1 . . . u k u 1 . . . u k ⊗ w.
Thus, the expressions of π 1 (w) and π * 1 (w) follow immediatly.

2. Since exp and log are mutually inverse then, by the previous results, one has

w∈Y * w ⊗ w = k≥0 1 k! w∈Y + w ⊗ π 1 (w) k = k≥0 1 k! u1,...,u k ∈Y + (u 1 . . . u k ) ⊗ (π 1 (u 1 ) . . . π 1 (u k )) = w∈Y + w ⊗ k≥1 1 k! u1,...,u k ∈Y + w | u 1 . . . u k π 1 (u 1 ) . . . π 1 (u k ) .
In the same way,

w∈Y * w ⊗ w = k≥0 1 k! u1,...,u k ∈Y + (π * 1 (u 1 ) . . . π * 1 (u k )) ⊗ (u 1 . . . u k ) = w∈Y + k≥0 1 k! u1,...,u k ∈Y + w | u 1 . . . u k π * 1 (u 1 ) . . . π * 1 (u k ) ⊗ w.
It follows then the expected result.

Pair of bases in duality

Definition 16. Let {Σ l } l∈LynY be the family of Lie Q Y obtained as follows

Σ y k = π 1 (y k ) for k ≥ 1, Σ l = [Σ s , Σ r ] for l ∈ LynX, standard factorization of l = (s, r),
and the family {Σ w } w∈Y * of U(Lie Q Y ) (viewed as a Q-module) obtained as follows

Σ l = 1 for l = 1 Y * , Σ w = Σ i1 l1 . . . Σ i k l k for w = l i1 1 . . . l i k k , l 1 > . . . > l k , l 1 . . . , l k ∈ LynY.
Let { Σw } w∈Y * be the family of the quasi-shuffle algebra (viewed as a Q-module) obtained by duality with {Σ w } w∈Y * :

∀u, v ∈ Y * , Σv | Σ u = δ u,v .
Proposition 19.

1. For l ∈ LynY , the polynomial Σ l is upper triangular :

Σ l = l + v>w,(v)=(l) c v v.
2. The families {Σ w } w∈Y * and { Σw } w∈Y * are upper and lower triangular respectively. On other words, for any w ∈ Y + , one has

Σ w = w + v>w,(v)=(w) c v v and Σw = w + v<w,(v)=(w) d v v,
where, for any y k ∈ Y and w ∈ Y * , (w) denotes the degree of w and

(y k ) = deg(y k ) = k.
Proof.

1. Let us prove it by induction on the length of l :

• The result is immediat for l ∈ Y .
• The result is suppose verified for any l ∈ LynY ∩ Y k and 0 ≤ k ≤ N .

• At N + 1, by the standard factorization (l 1 , l 2 ) of l, one has, by definition, Σ l = [Σ l1 , Σ l2 ] and l 2 l 1 > l 1 l 2 = l. By induction hypothesis,

Σ l1 = l 1 + v>l 1 (v)=(l 1 ) c v v and Σ l2 = l 2 + u>l 2 (v)=(l 2 ) d u u, ⇒ Σ l = l + w>l (w)=(l)
e w w, getting e w 's from c v 's and d u 's. Actually, the Lie bracket gives

Σ l = [l 1 , l 2 ] + u>l 2 (v)=(l 2 ) d u l 1 u + v>l 1 ,u>l 2 (v)=(l 1 ),(u)=(l 2 ) c v d u vu - v>l 1 (v)=(l 1 ) c v l 2 v - v>l 1 ,u>l 2 (v)=(l 2 ),(u)=(l 1 ) c v d u uv = [l 1 , l 2 ] + u>l 1 l 2 (v)=(l 1 l 2 ) d ′ u u + vu>l 1 l 2 (vu)=(l 1 l 2 ) c v d u vu - v>l 2 l 1 (v)=(l 2 l 1 ) c ′ v v - uv>l 2 l 1 (uv)=(l 2 l 1 ) c v d u uv = [l 1 , l 2 ] + u>l (v)=(l) d ′ u u + vu>l (vu)=(l) c v d u vu - v>l 2 l 1 >l (v)=(l) c ′ v v - uv>l 2 l 1 >l (uv)=(l) c v d u uv.
Hence, the conclusion follows.

2. Let w = l 1 . . . l k , with l 1 > . . . > l k and l 1 , . . . , l k ∈ LynY . By (ii),

Σ li = l i + v>l i (v)=(l i ) c i,v v and Σ w = l 1 . . . l k + u>w (v)=(w) d u u,
where the d u 's are obtained from the c i,v 's. Hence, the family {Σ w } w∈Y * is upper triangular and, by duality, the family { Σw } w∈Y * is lower triangular.

Theorem 20.

1. The family {Σ l } l∈LynY forms a basis of the free Lie algebra.

2. The family {Σ w } w∈Y * forms a basis of the free associative algebra Q Y . Proof.

1. Since Σ 1 Y * = 1 then Σ1 Y * = 1. 2. Let u = u 1 . . . u n = l i1 1 . . . l i k k , v = v 1 . . . v m = h j1 1 . . . h jp p with l 1 . . . , l k , h 1 , . . . , h p , u 1 , . . . , u n , v 1 , . . . , v m ∈ LynY, l 1 > . . . > l k , h 1 > . . . > h p , u 1 ≥ . . . ≥ u n , v 1 ≥ . . . ≥ v m and i 1 + . . . + i k = n, j 1 + . . . + j p = m. Hence, if m ≥ 2 (resp. n ≥ 2) then v / ∈ LynY (resp. u / ∈ LynY ). Since Σu1 . . . Σun | n i=1 Σ ui = Σu1 ⊗ . . . ⊗ Σun | ∆ (n-
d v v = w∈Y * Σw | u Σ bw - w∈Y * v>bw,(v)=(bw) d v w ′ ∈Y * Σw ′ | v Σ w ′ (by decomposing v in PBW-Lyndon basis) = w∈Y * Σw | u Σ bw - w∈Y * w ′ ∈Y * v>bw,(v)=(bw) d v Σw ′ | v Σ w ′ = bw∈LynY Σw | u Σ bw - w ′ ∈LynY w∈Y * v>bw,(v)=(bw) d v Σw ′ | v Σ w ′
+ sum of decreasinge products, of length ≥ 2, of Lie polynomials.

After splitting these two sums on two disjoint supports, one has

• for any bw = l i1 1 . . . l in n / ∈ LynY (n ≥ 2) with l 1 , . . . , l n ∈ LynY verifying l 1 > . . . > l n and Σ aw = Σ in l1 . . . Σ in ln . • for any w ′ = λ j1 1 . . . λ jm m / ∈ LynY (m ≥ 2) with λ 1 , . . . , λ m ∈ LynY verifying λ 1 > . . . > λ m and Σ w ′ = Σ j1 λ1 . . . Σ jm λm .
In the second sum, since each word v is great than the Lyndon word bw then the Lie polynomial Σ bw does not appear in the decomposition, in the PBW-Lyndon basis, of v. More precisely, (see Proposition 19)

Σw ′ = w ′ + w ′ >v ′ ,(w ′ )=(v ′ ) e v ′ v ′ with e v ′ ≥ 0, ⇒ Σw ′ | v = w ′ | v + w ′ >v ′ ,(w ′ )=(v ′ ) e v ′ v ′ | v .
In particular (for w ′ = bw ∈ LynY ), the coefficient of the Lie polynomial Σ bw in the decomposition of v (> bw) is vanishing : 3. The family LynY forms a transcendence basis 30 of the quasi-shuffle algebra and the family of proper polynomials of rational positive coefficients defined by, for any w = l i1 1 . . . l i k k with l 1 > . . . > l k and l 1 , . . . , l k ∈ LynY ,

Σbw | v = bw | v + v>bw>v ′ ,(bw)=(v ′ ) e v ′ v ′ | v = 0.
χ w = l i1 1 . . . l i k k i 1 ! . . . i k !
forms a basis of the quasi-shuffle algebra.

4. Let {ξ w } w∈Y * be the basis of the envelopping algebra U(Lie Q X ) obtained by duality with the basis {χ w } w∈Y * :

∀u, v ∈ Y * , χ v | ξ u = δ u,v .
Then the family {ξ l } l∈LynY forms a basis of the free Lie algebra Lie Q Y .

Proof.

1. The proof can be done by induction on the length of w using the fact that the product conserve the property, l'homogenity and rational positivity of the coefficients.

2. Expressing w in the basis { Σw } w∈Y * of the quasi-shuffle algebra and then in the basis {Σ w } w∈Y * of the envelopping algebra, we obtain successively

w∈Y * w ⊗ w = w∈Y * u∈X * Σ u | w Σu ⊗ w = u∈Y * Σu ⊗ w∈X * Σ u | w w = u∈Y * Σu ⊗ Σ u = l 1 >...>l k i 1 ,...,i k ≥1 Σ⊔⊔ i1 l1 . . . Σ⊔⊔ i k l k i 1 ! . . . i k ! ⊗ Σ i1 l1 . . . Σ i k l k = ց l∈LynY i≥0 Σ i l i! ⊗ Σ i l = ց l∈LynY exp( Σl ⊗ Σ l ).
3. For w = l i1 1 . . . l i k k with l 1 , . . . , l k ∈ LynY and l 1 > . . . > l k , by Proposition 19, the polynomial of rational positive coefficients Σw is lower triangular :

Σw = Σ i1 l1 . . . Σ i k l k i 1 ! . . . i k ! = w + v<w,(v)=(w) c v v.
In particular, for any l j ∈ LynY , Σlj is lower triangular :

Σlj = l j + v<lj ,(v)=(lj) c v v.
Hence, Σw = χ w +χ ′ w , where χ ′ w is a proper polynomial of Q Y of rational positive coefficients. We deduce then the support of χ w contains words which are less than w and χ w | w = 1. Thus, the proper polynomial χ w of rational positive coefficients is lower triangular :

χ w = w + v<w,(v)=(w) c v v, ⇒ ∀l ∈ LynY, χ l = l + v<l,(v)=(l) c v v.
It follows then expected results. 4. By duality, for w ∈ Y * , the proper polynomial ξ w is upper triangular. In particular, for any l ∈ LynY , the proper polynomial ξ l is upper triangular :

ξ l = l + v>l,(v)=(l) d v v.
Hence, the family {ξ l } l∈LynY is free and its elements verify an analogous of the generalized criterion of Friedrichs :

• for w ∈ LynY , one has χ w | ξ l = δ w,l ,

• for w / ∈ LynY , w = l 1 . . . l n with l 1 , . . . , l n ∈ LynY and l 1 > . . . > l n , one has χ w | ξ l = χ l1 . . . χ ln | ξ l = 0.

Moreover, the polynomials ξ l 's are primitive : by Corollary 16(3), one has

∆ (ξ l ) = u,v∈Y * u v | ξ l u ⊗ v = u∈Y + u 1 Y * | ξ l u ⊗ 1 Y * + v∈Y + 1 Y * v | ξ l 1 Y * ⊗ v + u,v∈Y + u v | ξ l u ⊗ v + 1 Y * 1 Y * | ξ l 1 Y * ⊗ 1 Y * = ξ l ⊗ 1 Y * + 1 Y * ⊗ ξ l .
Because, after decomposing u and v on the basis {χ l } l∈LynY and by the previous criterion, the third term is vanishing. The last one is also vanishing since the ξ l 's are proper. By a theorem of Viennot, we obtain then the expected result.

Proof of Theorem 2

Applying the tensor product of isomorphisms H ⊗ Id (Proposition 1) on the diagonal series (Corollary 16(ii)), the infinite factorization, by Lyndon words, of the noncommutative generating series of harmonic sums follows 31 :

H(N ) = w∈Y * H w (N ) w = ց l∈LynY exp(HΣ l (N ) Σ l ). ( 111
)

Annexe B : differential realization

To facilitate reading, the following results are placed in this Annex which can be skipped by readers already familiar with the techniques developed by Fliess (and adapted by us for studies in this paper).

6.1 Polysystem and convergence criterion

Serial estimates from above

Here, generalizing a little, K is supposed a C-algebra and a complete normed vector space equipped with a norm denoted by . . For any n ∈ N, X ≥n denotes the set of words over X of length greater than or equal to n. The set of formal power series (resp. polynomials) on X, is denoted by K X (resp. K X ).

Definition 17 ([25, 39]). Let ξ, χ be real positive functions over X * . Let S ∈ K X .

1. S will be said ξ-exponentially bounded from above if it verifies

∃K ∈ R + , ∃n ∈ N, ∀w ∈ X ≥n , S | w ≤ Kξ(w)/| w |!.
We denote by K ξ-em X the set of formal power series in K X which are ξ-exponentially bounded from above.

2. S verifies the χ-growth condition if it satisfies

∃K ∈ R + , ∃n ∈ N, ∀w ∈ X ≥n , S | w ≤ Kχ(w) | w |!.
We denote by K χ-gc X the set of formal power series in K X verifying the χ-growth condition. 31 This proof omitted in previous versions uses mainly the results presented in this annexe that have not been published earlier but have already been presented at various workshops. It is an analogous way to obtain the infinite factorization, by Lyndon words over the alphabet X, of the noncommutative generating series of polylogarithms (see Theorem 3) by applying the tensor product of isomorphisms Li ⊗Id (see Proposition 1) on the diagonal series, over X.

Lemma 24. We have

R = w∈X * | w |! w ⇒ R ⊔⊔ 2 | w = u,v∈X * supp(u ⊔⊔ v)∋w | u |! | v |! ≤ 2 |w| | w |!. Proof. One has u,v∈X * supp(u ⊔⊔ v)∋w | u |! | v |! = |w| k=0 |u|=k,|v|=|w|-k supp(u ⊔⊔ v)∋w k!(| w | -k)! = |w| k=0 | w | k k!(| w | -k)! = |w| k=0 | w |! = (1+ | w |) | w |!.
By induction on the length of w, one has 1 + |w| ≤ 2 |w| . It follows the expected result.

Proposition 20. Let S 1 and S 2 verifying the growth condition. Then S 1 + S 2 and S 1 ⊔⊔ S 2 also verifies the growth condition.

Proof. The proof for

S 1 + S 2 is immediate. Next, since S i | w ≤ K i χ i (w) | w |!, for i = 1 or 2 and for w ∈ X * , then 32 S 1 ⊔⊔ S 2 | w = supp(u ⊔⊔ v)∋w S 1 | u S 2 | v , ⇒ S 1 ⊔⊔ S 2 | w ≤ K 1 K 2 u,v∈X * supp(u ⊔⊔ v)∋w (χ 1 (u) | u |!)(χ 2 (v) | v |!). Let K = K 1 K 2 and let χ be a real positive function over X * such that ∀w ∈ X * , χ(w) = max{χ 1 (u)χ 2 (v) | u, v ∈ X * and supp(u ⊔⊔ v) ∋ w}.
With the notations in Lemma 24, we get

S 1 ⊔⊔ S 2 | w ≤ Kχ(w) R ⊔⊔ 2 | w .
Hence, S 1 ⊔⊔ S 2 verifies the χ ′ -growth condition with χ ′ defined as χ ′ (w) = 2 |w| χ(w).

Definition 18 ([25, 39]). Let ξ be a real positive function defined over X * , S will be said ξ-exponentially continuous if it is continuous over K ξ-em X . The set of formal power series which are ξ-exponentially continuous is denoted by K ξ-ec X .

Lemma 25 ([25, 39]). For any real positive function ξ defined over X * , we have K X ⊂ K ξ-ec X . Otherwise, for ξ = 0, we get K X = K 0-ec X . Hence, any polynomial is 0-exponentially continuous.

Proposition 21 ([25, 39]). Let ξ, χ be a real positive functions over X * and let P ∈ K X .

1. Let S ∈ K ξ-em X . The right residual of S by P belongs to K ξ-em X .

2. Let R ∈ K χ-gc X . The concatenation SR belongs to K χ-gc X .

Proof.

1. Since S ∈ K ξ-em X then ∃K ∈ R + , ∃n ∈ N, ∀w ∈ X ≥n , S | w ≤ Kξ(w)/| w |!.
If u ∈ supp(P ) := {w ∈ X * | P | w = 0} then, for any w ∈ X * , one has S ⊲ u | w = S | uw and S ⊲ u belongs to K ξ-em X :

∃K ∈ R + , ∃n ∈ N, ∀w ∈ X ≥n , S ⊲ u | w ≤ [Kξ(u)]ξ(w)/| w |!. It follows then S ⊲ P is K ξ-em X by taking K 1 = K max u∈supp(P ) ξ(u). 2. Since R ∈ K χ-gc X then ∃K ∈ R + , ∃n ∈ N, ∀w ∈ X ≥n , S | w ≤ Kχ(w) | w |!. Let v ∈ supp(P ) such that v = ǫ. Since, for any w ∈ X * , Rv belongs to K χ-gc X and one has Rv | w = R | v ⊳ w : ∃K ∈ R + , ∃n ∈ N, ∀w ∈ X ≥n , R | v ⊳ w ≤ Kχ(v ⊳ w)(| w | -| v |)! ≤ K | w | χ(w)/χ(v).
Note that if v ⊳w = 0 then Rv | w = 0 and the previous conclusion holds.

It follows then RP is K χ-gc X by taking K 2 = K min v∈supp(P ) χ(v) -1 .

Proposition 22 ([25, 39]). Two real positive morphisms over X * , ξ and χ are assumed to verify the condition x∈X χ(x)ξ(x) < 1.

Then for any F ∈ K χ-gc X , F is continuous over K ξ-em X .

Proof. If ξ, χ verify the upper bound condition then the following power series

w∈X * χ(w)ξ(w) = x∈X χ(x)ξ(x) *
is well defined. If F ∈ K χ-gc X and C ∈ K ξ-em X then there exists K i ∈ R + and n i ∈ N such that for any w ∈ X ≥ni , i = 1, 2, one has

F | w ≤ K 1 χ(w) | w |! and C | w ≤ K 2 ξ(w)/| w |!. Hence, ∀w ∈ X * , | w |≥ max{n 1 , n 2 }, F |w C|w ≤ K 1 K 2 χ(w)ξ(w), ⇒ w∈X * F |w C|w ≤ K 1 K 2 w∈X * χ(w)ξ(w) = K 1 K 2 x∈X χ(x)ξ(x) * .

Upper bounds à la Cauchy

Let q 1 , . . . , q n be commutative indeterminates over C. The algebra of formal power series (resp. polynomials) over {q 1 , . . . , q n } with coefficients in C is denoted by C[[q 1 , . . . , q n ]] (resp. C[q 1 , . . . , q n ]).

Definition 19 ([25, 39]). Let

f = i1,...,in≥0 f i1,...,in q i1 1 . . . q in n ∈ C[[q 1 , . . . , q n ]].
We set

E(f ) := {ρ ∈ R n + : ∃C f ∈ R + s.t. ∀i 1 , . . . , i n ≥ 0, | f i1,...,in | ρ i1 1 . . . ρ in n ≤ C f } Ȇ(f ) : interior of E(f ) in R n . CV(f ) := {q ∈ C n : (| q 1 |, . . . , | q n |) ∈ Ȇ(f )} : convergence domain of f.
The power series f is to be said convergent if CV(f ) = ∅. Let U be an open domain in C n and let q ∈ C n . The power series f is to be said convergent on q (resp. over U) if q ∈ CV(f ) (resp. U ⊂ CV(f )). We set

C cv [[q 1 , . . . , q n ]] = {f ∈ C[[q 1 , . . . , q n ]] : CV(f ) = ∅}.
Let q ∈ CV(f ). There exists the constants C f , ρ and ρ such that | q 1 |< ρ < ρ, . . . , | q n |< ρ < ρ and, for i

1 . . . , i n ≥ 0, | f i1,...,in | ρ i1 1 . . . ρ in n ≤ C f . The convergence modulus of f at q is (C f , ρ, ρ). Suppose that CV(f ) = ∅ and let q ∈ CV(f ). If (C f , ρ, ρ) is a convergence modulus of f at q then | f i1,...,in q i1 1 . . . q in n |≤ C f (ρ 1 /ρ 1 ) i1 . . . (ρ 1 /ρ 1 ) in .
Hence, at q, the power series f is majored termwise by

C f m k=0 1 - ρk ρ k -1 . (112)
Hence, f is uniformly absolutely convergent in {q ∈ C n :| q 1 |< ρ, . . . , | q n |< ρ} which is an open domain in C n . Thus, CV(f ) is an open domain in C n . Since the partial derivation of order j 1 , . . . , j n ≥ 0 of f is estimated by

D j1 1 . . . D jn n f ≤ C f ∂ j1+...+jn ∂ ρj1+...+jn m k=0 1 - ρk ρ k -1 . ( 113
) Proposition 23 ([25]). We have CV(f ) ⊂ CV(D j1 1 . . . D jn n f ). Let f ∈ C cv [[q 1 , . . . , q n ]
] and let {A i } i=0,1 be a polysystem defined as follows

A i (q) = n j=1 A j i (q) ∂ ∂q j , with A j i (q) ∈ C cv [[q 1 , . . . , q n ]], j = 1, . . . , n. (114)
Lemma 26 ([20]). For i = 0, 1 and j = 1, .., n, one has A i • q j = A j i (q). Thus,

∀i = 0, 1, A i (q) = n j=1 (A i • q j ) ∂ ∂q j . Let (ρ, ρ, C f ), {(ρ, ρ, C i )} i=0,1 be respectively the convergence modulus at q ∈ CV(f ) i=0,1 j=1,..,n CV(A j i ) (115)
of f and {A j i } j=1,..,n . Let us consider the following monoid morphisms

A(ǫ) = identity and C(ǫ) = 1, (116) ∀w = vx i , x i ∈ X, v ∈ X * , A(w) = A(v)A i and C(w) = C(v)C i .(117)
Lemma 27 ([19]). For any word w, A(w) is continuous over C cv [[q 1 , . . . , q n ]] and, for any f, g ∈ C cv [[q 1 , . . . , q n ]], one has

A(w) • (f g) = u,v∈X * u ⊔⊔ v | w (A(u) • f )(A(v) • g).
These notations are extended, by linearity, to K X and we will denote A(w) • f |q the evaluation of A(w) • f at q. Definition 20 ([19]). Let f ∈ C cv [[q 1 , . . . , q n ]]. The generating series of the polysystem {A i } i=0,1 and of the observation f is given by

σf := w∈X * A(w) • f w ∈ C cv [[q 1 , . . . , q n ]] X . σf |q := w∈X * A(w) • f |q w ∈ C X .
The last generating series is called Fliess generating series of the polysystem {A i } i=0,1 and of the observation f at q. Lemma 28 ([19]). Let {A i } i=0,1 be a polysystem. Then, the map σ : (C cv [[q 1 , . . . , q n ]], .) -→ (C cv [[q 1 , . . . , q n ]] X , ⊔⊔ ), is an algebra morphism, i.e. for any f, g ∈ C cv [[q 1 , . . . , q n ]] and µ, ν ∈ C, one has σ(νf + µh) = νσf + µσg and σ(f g) = σf ⊔⊔ σg. Lemma 29 ([20]). Let {A i } i=0,1 be a polysystem and f ∈ C cv [[q 1 , . . . , q n ]]. Then

∀x i ∈ X, σ(A i • f ) = x i ⊳ σf ∈ C cv [[q 1 , . . . , q n ]] X ∀w ∈ X * , σ(A(w) • f ) = w ⊳ σf ∈ C cv [[q 1 , . . . , q n ]] X .
Lemma 30 ([25]). Let τ = min 1≤k≤n ρ k and r = max 1≤k≤n ρk /ρ k . We have

A(w) • f ≤ C f (n + 1) (1 -r) n C(w) | w |! n+|w|-1 |w| n τ (1 -r) n+1 |w| ≤ C f (n + 1) (1 -r) n C(w) n τ (1 -r) n+1 |w| | w |!. Theorem 22 ([25]). Let K = C f (n + 1)(1 -r) -n
and let χ be the real positive function defined over X * by ∀i = 0, 1, χ(x i ) = C i n τ (1r) (n+1) .

Then the generating series σf of the polysystem {A i } i=0,1 and of the observation f satisfies the χ-growth condition.

It is the same for the Fliess generating series σf |q of the polysystem {A i } i=0,1 and of the observation f at q.

Polysystems and nonlinear differential equation

and the following nonlinear dynamical system 34    y(z) = f (q(z)), q(z) = A 0 (q) u 0 (z) + A 1 (q) u 1 (z), q(z 0 ) = q 0 , (119) 33 These singular inputs are not included in the studies of Fliess motivated, in particular, by the renormalization of y(z) at +∞ [START_REF] Fliess | Fonctionnelles causales non linéaires et indéterminées non commutatives[END_REF][START_REF] Fliess | Réalisation locale des systèmes non linéaires, algèbres de Lie filtrées transitives et séries génératrices non commutatives[END_REF]. 34 Any differential equation with singularities in {a, b, c}, via homographic transformation (za)(cb)(zb) -1 (ca) -1 , can be changed into a differential equation with singularities in {0, 1, +∞} (the singularities of homographic transformations belonging to the group G).

where, the state q = (q 1 , . . . , q n ) belongs to the complex analytic manifold of dimension n, q 0 is the initial state, the observation f belongs to C cv [[q 1 , . . . , q n ]] and {A i } i=0,1 is the polysystem defined on (114). • an integer d,

• a power series f ∈ K[[q 1 , . . . , qd ]],

• a homomorphism A from X * maps to the algebra of differential operators generated by

A(x i ) = d j=1
A j i (q 1 , . . . , qd ) ∂ ∂ qj , A j i (q 1 , . . . , qd ) ∈ K[[q 1 , . . . For any j = 1, .., d, we put

T j = w∈X * ∂(A(w) • f ) ∂ qj w ⇐⇒ ∀w ∈ X * , T j | w = ∂(A(w) • f ) ∂ qj .
Firstly, by Theorem 22, the generating series σf verifies the growth condition. Secondly, for any Π ∈ Lie K X and for any w ∈ X * , one has Proof. Assume their exists j ∈ [1, .., d] such that S j does not verify the growth condition. Since S ∈ Ann ⊥ (S) then using the decomposition of S on S 1 , . . . , S d , one obtains a contradiction with the fact that S verifies the growth condition.

Conservely, using Proposition 20, we get the expected results.

Theorem 24 ([20]). The formal power series S ∈ K X is differentially produced if and only if its Lie rank is finite and if it verifies the χ-growth condition.

Thus, these yield also identities on polyzêtas at arbitrary weight [START_REF] Ngoc | Finite polyzêtas, Poly-Bernoulli numbers, identities of polyzêtas and noncommutative rational power series[END_REF].

  As already shown by Drinfel'd, the equation (DE) admits, on the simply connected domainC -(] -∞, 0] ∪ [1, +∞[), two specific solutions G 0 (z) z 0 exp[x 0 log(z)] and G 1 (z) z 1 exp[-x 1 log(1z)].

1 . 2 ) 8 )

 128 Li r1,...,r k (z) = n1>...>n k >0 z n1 n r1 1 . . . n r k k (on the asymptotic expansion of polylogarithms, at z = 1 and in the comparison scale {(1z) a log b (1z)} a∈Z,b∈N , and the same action on the asymptotic expansions, at +∞ and in the comparison scales {n a log b (n)} a∈Z,b∈N and {n a H b 1 (n)} a∈Z,b∈N , of the harmonic sums (recalled in Section 2.1.1) H r1,...,r k (N ) = N n1>...>n k >0 1 n r1 1 . . . n r k k .

  w∈Y * Li w (z) w and H(N ) = w∈Y * H w (N ) w.

2. 3 . 1

 31 Residual calculus and representative series Definition 4. Let S ∈ Q X and let P ∈ Q X .

Lemma 7 .

 7 Let H be a monoid containing {e t x } t∈C x∈X . Let S ∈ C cont X being indiscernable over H. Then for any x ∈ X, x ⊳ S and S ⊲ x belong to C cont X and they are indiscernable over H. Proof. Let us calculate x ⊳ S || Φ = S || Φx and S ⊲ x || Φ = S || xΦ . Since lim t→0 e t x -1 t = x and lim t→0 e t x -1 t = x then, for any Φ ∈ H, by uniform convergence, one has

3. 1 3 . 1 . 1

 1311 Generalized Euler constants and global regularization of polyzêtas Three regularizations of divergent polyzêtas Theorem 8 ([33]). Let ζ : (Q Y ,

σ

  Li w w = ց l∈LynX e σ Li Šl S l . (64) Since dσ Li xi = σd Li xi = ω i then by integrating the two memmbers, we obtain σ Li xi = Li xi +c xi , where c xi is a constant of integration. More generally, for any Lyndon word l

  0 and on the other, Φ | ǫ = Z ⊔⊔ | ǫ = 1, the result follows.Note also that if Z ⊂ A then dm(A) forms a group and with the notations of Corollary 6, we obtain Corollary 8. For any associator Φ = Z ⊔⊔ e C ∈ dm(A), let Ψ = B(y 1 )π Y Φ and letΨ ′ = B ′ (y 1 )π Y Φ. Then Ψ = B(y 1 )π Y Φ ⇐⇒ Ψ ′ = B ′ (y 1 )π Y Φ.Proof. Since Ψ is group like and since Φ | x 1 = Ψ ′ | y 1 = 0 and Ψ | y 1 = γ then, using the factorization by Lyndon words, we get the expected result.Lemma 10. Let Φ = Z ⊔⊔ e C ∈ dm(A) and let Ψ = B(y 1 )π Y (Z ⊔⊔ e C ). The local coordinates (of second kind) of Φ (resp. Ψ) are polynomials on {ζ ⊔⊔ ( Šl )} l∈LynX (resp. {ζ ( Σl )} l∈LynY ) of Z (resp. Z ′ ). While C describes Lie A X , these coordinates describe A[{ζ ⊔⊔ ( Šl )} l∈LynX ] (resp. A[{ζ ( Σl )} l∈LynY ]).Proof. Let Φ ∈ dm(A). By Corollary 7, there exists P ∈ Lie A X verifying e P | ǫ = 1, e P | x 0 = e P | x 1 = 0 such that Φ = Z ⊔⊔ e P . Using the factorization forms by Lyndon words, we get ց l∈LynX-X e φ( Šl ) S l = ց l∈LynX-X e ζ( Šl ) S l ց l∈LynX-X e p Šl S l . Expanding the Hausdorff product and identifying the local coordinates in the PBW-Lyndon basis there exists I l ⊂ {λ ∈ LynX -X s.t. |λ| ≤ |l|}, for l ∈ LynX -X, and the coefficients {p ′ Šu } u∈I l belonging to A such that φ( Šl ) = u∈I l p ′ Šu ζ( Šu ). This belongs to A[{ζ( Šl )} l∈LynX-X ] and holds for any P ∈ Lie A X . With the notations of Definition 9 and by Corollary 8, we get in particular Lemma 11. For any Φ ∈ dm(A), by identifying the local coordinates (of second kind) on two members of the identities Ψ = B(y 1 )π Y Φ, or equivalently of Ψ ′ = B ′ (y 1 )π Y Φ, we get polynomial relations, of coefficients in A, among generators of the A-algebra of convergent polyzêtas.

Lemma 20 .Lemma 21 .

 2021 Let Φ ∈ dm(A). Let us define the map ϕ : Z -→ A as follows ∀l ∈ L irr X, ϕ(ζ(l)) := φ(l).Then ϕ is an algebra homomorphism and {ϕ(ζ(l))} l∈Lirr X are generators of A.Thus, for any θ ∈ Z there exist the coefficients {α l1,...,ln } n∈N l1,...,ln∈Lirr X in A such that (see Proposition 17 and Lemme 20)ϕ(θ) =n≥0 l1,...,ln∈Lirr X α l1,...,ln ϕ(ζ(l 1 )) . . . ϕ(ζ(l n )). (86) In particular, since for any w ∈ X * , ζ ⊔⊔ (w) belongs to Z (see Corollary 3) then ϕ(ζ ⊔⊔ (w)) is well defined and ϕ(ζ ⊔⊔ (w)) can be expressed as polynomial on convergent polyzêtas with coefficients in A : With the notations in Lemma 20, one has ∀w ∈ X * , ϕ(ζ ⊔⊔ (w)) = u,v∈X * uv=w e C | v ζ ⊔⊔ (u).

Corollary 12 .Corollary 14 .

 1214 If γ / ∈ A then it is transcendental over the A-algebra generated by the convergent polyzêtas. Or equivalently, by contraposition, Corollary 13. If there exists a polynomial relation with coefficients in A among the Euler constant, γ, and the convergent polyzêtas then γ ∈ A. Therefore, If the Euler constant, γ, does not belong to A then γ is not algebraic over A. Remark 3.

  and each monomial in (110) is then of different weight. By consequence, Corollary 15. Any (A-irreducible) polyzêta θ is a transcendental over Q.

Lemma 23 . 1 . 2 . 3 .

 23123 Let S 1 , . . . , S n be proper formal power series in Q Y . Let P 1 , . . . , P m be primitive elements 29 in Q Y , for the co-product . If n > m then S 1 . . . S n | P 1 . . . P m = 0. If n = m then S 1 . . . S n | P 1 . . . P n = σ∈Sn n i=1 S i | P σ(i) . If n < m then, by considering the language M over A = {P 1 , . . . , P m } M = {w ∈ A * |w = P j1 . . . P j |w| , j 1 < . . . < j |w| , |w| ≥ 1} and the morphism µ : Q A -→ Q Y , one has : S 1 . . . S n | P 1 . . . P m = w 1 ,...,wm∈M |w 1 |+...+|wn |=m ∀i,j=1,..m,alp(w i )∩alp(w j )=∅ n i=1 S i | µ(w i ) .

3 .Σ i k l k i 1 ! 3 .

 313 The family { Σw } w∈Y * generate freely the quasi-shuffle algebra. 4. The family { Σl } l∈LynY forms a transcendence basis of the quasi-shuffle algebra. Proof. The family {Σ l } l∈LynY of upper triangular polynomials is free and then, by a theorem of Viennot, we get the first result. The second is a direct consequence of the Poincaré-Birkhoff-Witt theorem. By the Cartier-Quillen-Milnor-Moore theorem, we get the third one and the last one is also obtained as consequence of the constructions of the families { Σl } l∈LynY and { Σw } w∈Y * of lower triangular polynomials. Now, let us clarify the basis { Σw } w∈Y * and then the transcendence basis { Σl } l∈LynY of the quasi-shuffle algebra (Q Y , ) as follows Theorem 21. We have 1. For w = 1 Y * , Σw = 1. 2. For any w = l i1 1 . . . l i k k , with l 1 , . . . , l k ∈ LynY and l 1 > . . . > l k , . . . i k ! . For any y ∈ Y, Σy = π * 1 (y).

4 .

 4 For any l = yu ∈ LynY -Y , Σl = y Σu .

Thus, by identifying

  the coefficients in these two expressions of Lyndon word bu, one has Σaw | bu = δ a,b Σw | u . In other words, Σbw = b Σw . Corollary 16.1. For w ∈ Y + , the polynomial Σw is proper and homogenous of degree |w|, for deg(y i ) = i, and of rational positive coefficients. 2. w∈Y * w ⊗ w = w∈Y * Σw ⊗ Σ w = ց l∈LynY exp( Σl ⊗ Σ l ).

6. 2 . 1

 21 Nonlinear differential equation (with three singularities) Let us consider the following singular inputs 33 u 0 (z) := z -1 and u 1 (z) := (1z) -1 ,

Definition 21 (

 21 [START_REF] Ngoc | Input/Output Behaviour of Nonlinear Control Systems : Rational Approximations, Nilpotent structural Approximations[END_REF]). The following power series is called transport operator of the polysystem {A i } i=0,1 and of the observation fT := w∈X * α z z0 (w) A(w).By the factorization of the monoid by Lyndon words, we have[START_REF] Ngoc | Input/Output Behaviour of Nonlinear Control Systems : Rational Approximations, Nilpotent structural Approximations[END_REF] T = (α z z0 ⊗ A)w∈X * w ⊗ w = l∈LynX exp[α z z0 (S l ) A( Šl )]. (120)Let us consider again the Chen generating series S z0 z given in (51) of the diffferential forms involed in (DE) of Example 1, i.e. ω 0 (z) = u 0 (z) dz and ω 1 (z) = u 1 (z) dz, verifying the upper bound conditions given on (56).

6. 2 . 2

 22 Asymptotic behaviour of the successive differentiation of the output via extended Fliess fundamental formulaTheorem 23 ([39]). The Fliess fundamental formula can be extended as followsy(z) = T • f |q 0 = w∈X * S z0 z | w A(w) • f |q 0 | w = σf |q 0 || S z0 z .By the factorization of the Lie exponential series L, it follows the expansions of the output y of nonlinear dynamical system with singular inputs, Corollary 17([39]).y(z) = w∈X * g w (z) A(w) • f |q0 , = k≥0 n1,...,n k ≥0 g x n 1 0 x1...x n k 0 x1 (z) ad n1 A0 A 1 . . . ad n k A0 A 1 e log zA0 • f |q0 , = l∈LynX exp g S l (z) A( Šl ) • f |q0 , = exp w∈X * g w (z) A(π 1 (w)) • f |q 0 ,where, for any word w in X * , g w belongs to the polylogarithm algebra.Since S z0 z = L(z)L(z 0 ) -1 and since σf |q 0 and L(z 0 ) -1 are invariant by∂ = d/dz then ∂ l y(z) = σf |q 0 || ∂ l S z0 z = σf |q 0 || ∂ l L(z)L(z 0 ) -1 , for l ≥ 0.With the notations of Proposition 3, we get ∂ l y(z) = σf |q 0 || [P l (z)L(z)]L(z 0 ) -1 = σf |q 0 ⊲ P l (z) || L(z)L(z 0 ) -1 . (121)

  , qd ]], j = 1, . . . , d, such that, for any w ∈ X * , S | w = A(w) • f |0 . The couple (A, f ) is called differential representation of S of dimension d. Proposition 24 ([46]). Let S ∈ K X . If S is differentially produced then it verifies the growth condition and its Lie rank is finite. Proof. Let (A, f ) be a differential representation of S of dimension d. Then, by the notations of Definition 20, we get σf |0 = S = w∈X * (A(w) • f ) |0 w.

  σf ⊲ Π | w = σf | Πw = A(Πw) • f = A(Π) • (A(w) • f ).Since A(Π) is a derivation over K[[q 1 , . . . , qd ]] :A(Π) = d j=1 (A(Π) • qj ) ∂ ∂ qj , ⇒ A(Π) • (A(w) • f ) = d j=1 (A(Π) • qj ) ∂(A(w) • f ) ∂ qj then we deduce that ∀w ∈ X * , σf ⊲ Π | w = d j=1 (A(Π) • qj ) T j | w , ⇐⇒ σf ⊲ Π = d j=1 (A(Π) • qj ) T jThat means σf ⊲ Π is K-linear combination of {T j } j=1,..,d and the dimension of the vector space span{σf ⊲ Π | Π ∈ Lie K X } is less than or equal to d. 6.3.2 Fliess' local realization theorem Proposition 25 ([46]). Let S ∈ K X such that its Lie rank equals d. Then there exists a basis S 1 , . . . , S d ∈ K X of (Ann ⊥ (S), ⊔⊔ ) ∼ = (K[[S 1 , . . . , S d ]], ⊔⊔ ) such that the S i 's are proper and for any R ∈ Ann ⊥ (S), one has R = i1,...,i d ≥0 r i1,...,in i 1 ! . . . i d ! S ⊔⊔ i1 1 ⊔⊔ . . . ⊔⊔ S ⊔⊔ i d d , where the coefficients {r i1,...,i d } i1,...,i d ≥0 belong to K and r 0,...,0 = R | ǫ . Proof. By Lemma 31, a such basis exists. More precisely, since the Lie rank of S is d then there exists P 1 , . . . , P d ∈ Lie K X such that S ⊲ P 1 , . . . , S ⊲ P d ∈ (K X , ⊔⊔ ) are K-linearly independent. By duality, their exists S 1 , . . . , S d ∈ (K X , ⊔⊔ ) such that ∀i, j = 1, ..d, S i | P j = δ i,j , and R = d i=1 exp(S i P i ). Expending this product, one obtains, via Poincaré-Birkhoff-Witt theorem, the expected expression for the coefficients r i1,...,i d = R | P i1 1 . . . P i d d . Hence, (Ann ⊥ (S), ⊔⊔ ) is generated by S 1 , . . . , S d . With the notations of Proposition 25, one has respectivelyCorollary 20. If S ∈ K[S 1 , . . . , S d ] then, for any i = 0, 1 and j = 1, .., d, one has x i ⊳ S ∈ Ann ⊥ (S) = K[S 1 , . . . , S d ].Corollary 21. The power series S verifies the growth condition if and only if, for any i = 1, ..d, S i also verifies the growth condition.

Proof.⊔⊔ i1 1 i 1 ! 1 i 1 !

 1111 By Proposition 24, one gets a direct proof. Conversely, since the Lie rank of S equals d then by Proposition 25, by putting σf |0 = S and, for any j = 1, .., d, σ qi = S i , 1. we choose the observation f as followsf (q 1 , . . . , qd ) = i1,...,i d ≥0 r i1,...,in i 1 ! . . . i d ! qi1 1 . . . qi d d ∈ K[[q 1 , . . . , qd ]], such that σf |0 (q 1 , . . . , qd ) = i1,...,i d ≥0 r i1,...,in i 1 ! . . . i d ! (σ q1 ) ⊔⊔ i1 ⊔⊔ . . . ⊔⊔ (σ qd ) ⊔⊔ i d , 3. Assume S ∈ Kǫ ⊕ x 0 K X x 1and S is a differentially produced. If there exists a basis S 1 , . . . , S d of (Ann ⊥ (S),⊔⊔ ) ∼ = (x 0 K X x 1 , ⊔⊔ ) such that S = i1,...,i d ≥0 r i1,...,in S ⊔⊔ . . . ⊔⊔ S ⊔⊔ i d d i d ! ∈ (K[S 1 , . . . , S d ], ⊔⊔ ). (122)We put Σ i := π Y S i , for i = 1, .., d and thenΣ := i1,...,i d ≥0 r i1,...,in π i1 . . . Σ i d d i d ! ∈ (K[Σ 1 , . . . , Σ d ], ). (123)It is a generalization of a Radford's theorem because[START_REF] Ngoc | Fonctions de Dirichlet d'ordre n et de paramètre t[END_REF][START_REF] Ngoc | Calcul symbolique non commutatif : aspects combinatoires des fonctions spéciales et des nombres spéciaux[END_REF] :• If S ∈ Q X then (122), (123) are decompositions on Radford bases. • If S is rational then these are noncommutative partial decompositions. In general one has π Y S = Σ but ζ(S i ) = ζ(Σ i ) and ζ(S) = ζ(Σ) = i1,...,i d ≥0 r i1,...,in ζ(S 1 ) i1 i 1 ! . . . ζ(S d ) i d i d ! .

  Indiscernability over a class of formal power series . . . . . . . . 2.3.1 Residual calculus and representative series . . . . . . . . . 2.3.2 Continuity and indiscernability . . . . . . . . . . . . . . . 3 Group of associators : polynomial relations among convergent polyzêtas and identification of local coordinates 3.1 Generalized Euler constants and global regularization of polyzêtas 3.1.1 Three regularizations of divergent polyzêtas . . . . . . . . 3.1.2 Identities of noncommutative generating series of polyzêtas 3.2 Action of differential Galois group of polylogarithms on their asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . .

	1 Introduction
	1.1 Drinfel'd associator and polyzêtas

1 Introduction 1.1 Drinfel'd associator and polyzêtas . . . . . . . . . . . . . . . . . . 1.2 Group of associators and regularized Chen generating series . . . 1.3 Global renormalization and global regularization . . . . . . . . . 2 Background : structures and analytical studies of harmonic sums and of polylogarithms 2.1 Structures of harmonic sums and of polylogarithms . . . . . . . . 2.1.1 Quasi-symmetric functions and harmonic sums . . . . . . 2.1.2 Iterated integrals and polylogarithms . . . . . . . . . . . . 2.2 Results à la Abel for generating series of harmonic sums and of polylogarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Generating series of harmonic sums and of polylogarithms 2.2.2 Asymptotic expansions by noncommutative generating series and regularized Chen generating series . . . . . . . . 2.3

4

  Concluding remarks : complete description of ker ζ and structure of polyzêtas For the same raison as already said in Remark 2(2), let us consider now 26 the commutative indeterminates t 1 , t 2 , t 3 , . . .. Let A be the Q-algebra obtained by specializing Q[t 1 , t 2 , t 3 , . . .] at t 1 = iπ :

  1) (Σ v1 . . . Σ vm ) then many cases occur : then, by decomposing u in the PBW-Lyndon basis and then by multiplying by b, we get on other hand,

	bu =	w∈Y *	Σw | u bΣ w
	=	w∈Y *	Σw | u Σ bw -	v>bw,(v)=(bw)

i.e., L is goup-like for the co-product ∆ ⊔⊔ : ∆ ⊔⊔ (L) = L ⊗ L.

Here, γ stands for the Euler constant γ = .57721566490153286060651209008240243 . . .

It can be shown that all minimal representations are isomorphics (see[START_REF] Berstel | Rational series and their languages[END_REF]).

A series S is said to be proper if S | ǫ = 0.

See[START_REF] Ngoc | Contribution au développement d'outils informatiques pour résoudre des problèmes d'automatique non linéaire[END_REF][START_REF] Ngoc | Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Inputs[END_REF], for a convergence criterion and an example of continuous generating series.

Here, we adapt this notion developped in[START_REF] Ngoc | Contribution au développement d'outils informatiques pour résoudre des problèmes d'automatique non linéaire[END_REF] via the residual calculus.

This double regularization is deduced from of the noncommutative generating series Z ⊔⊔ and Z on the definitions 1 and 2 (see the theorems 8 and 9).

See Corollary 22 of Annexe B.

See Corollary 22 of Annexe B.

We do not consider in any case A = Q as in previous versions.

i.e., for any i = 1, .., m, ∆ (P i ) = 1 ⊗ P i + P i ⊗ 1.

This result is an analogous of a Radford theorem (see[START_REF] Reutenauer | Free Lie Algebras[END_REF]). Thus the bases LynY and { Σl } l∈LynY belong to the class of Radford bases, i.e. the class of trancensdence bases, of the quasi-shuffle algebra, as well as the bases LynX and {S l } l∈LynX belong to the class of Radford bases of the shuffle algebra.

S 1 ⊔⊔ S 2 | w is the coefficient of the word w in the power series S 1 ⊔⊔ S 2 .

As in Corollary 6, we always get

Hence, roughly speaking, for the quasi-shuffle product, the symbolic regularization to T is also "equivalent" to the regularization to 0.

3. Again, as in Corollary 12, if T / ∈ A then T / ∈ Ā.

A contrario, as in Corollary 13, if there exists a polynomial relation with coefficients in A among T and convergent polyzêtas then T ∈ A.

Structure and arithmetical nature of polyzêtas

Once again, let us consider

(see Definition 10, Lemma 16, Definition 13). Then (A 1 , ⊔⊔ ) ∼ = (A 2 , ) [START_REF] Hoang | Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier[END_REF][START_REF] Hoang | De l'algèbre des ζ de Riemann multivariées l'algèbre des ζ de Hurwitz multivariées[END_REF].

Let us consider again the following algebra morphism (see Proposition 9)

Lemma 22. The image of the algebra morphism ζ is Z.

Let us make precise the structure of Z and the arithmetical nature of polyzêtas :

As consequences of the propositions 15, 16 and 17, by taking Φ = Z ⊔⊔ , we have

By Corollary 10, ker ζ is an ideal generated by the homogenous polynomials of degree ≥ 2. Hence, the quotien A 1 /R X or A 2 /R Ȳ (the source by the kernel of ζ) is graded [6] and it is isomorphic to Imζ. Therefore, by Lemma 22 and Proposition 17, we obtain respectively the following direct consequences

) then the second member is vanishing else, i.e. u = v, the second member equals 1 because the factorization by Lyndon words is unique.

(c) Case n < m. By Lemma 23(3), let us consider the following language over the alphabet A = {Σ v1 , . . . , Σ vm } :

and the morphism µ

Because in this product, on one hand, there exists at least one

Since for any aw

For z 0 = ε → 0 + , the asymptotic behaviour and the renormalization at z = 1 of ∂ l y(z) (or the asymptotic expansion and the renormalization of its Taylor coefficients at +∞) are deduced from Proposition 5 and extend a little bit the results of [START_REF] Ngoc | Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Inputs[END_REF] as follows Corollary 18. For any integer l,we have

Corollary 19. The differentiation of order l ∈ N of the output y of the dynamical system (119) is a C-combination of the elements g belonging to the polylogarithm algebra. If its ordinary Taylor expansion exists then the coefficients of this expansion belong to the algebra of harmonic sums and there exists algorithmically computable coefficients a i ∈ Z, b i ∈ N and c i belong to the C-algrebra generated by Z and by the Euler's γ constant, such that

n n→∞ i≥0 c i n ai log bi n.

Differential realization

It is immediate that Ann ⊥ (S) ∋ S and it follows that (see [START_REF] Fliess | Réalisation locale des systèmes non linéaires, algèbres de Lie filtrées transitives et séries génératrices non commutatives[END_REF][START_REF] Reutenauer | The local realisation of generating series of finite Lie rank[END_REF])

By Lemma 3, the residuals are derivations for shuffle product. Then, Lemma 32. Let S ∈ K X . Then :

For any

Definition 24 ([20]). The formal power series S ∈ K X is differentially produced if there exists 2. it follows that, for i = 0, 1 and for j = 1, .., d, the residuals x i ⊳ σ qj belongs to Ann ⊥ (σf |0 ) (see also Lemma 32), 3. since σf verifies the χ-growth condition then, by Corollary 21, the generating series σ qj and x i ⊳ σ qj (for i = 0, 1 and for j = 1, .., d) verify also the growth condition. We then take (see Lemma 29) ∀i = 0, 1, ∀j = 1, .., d, σA i j (q 1 , . . . , qd ) = x i ⊳ σ qj , by expressing σA i j on the basis {σ qi } i=1,..,d of Ann ⊥ (σf |0 ), 4. the homomorphism A is then determined as follows

where, for i = 0, 1, j = 1, .., d, A i j (q 1 , . . . , qd ) = A(x i ) • qj (see Lemma 26).

Thus, (A, f ) provides a differential representation 35 of dimension d of S.

Moreover, one also has the following Theorem 25 ([20]). Let S ∈ K X supposed to be a differentially produced formal power series. If (A, f ) and (A ′ , f ′ ) are two differential representations of dimension n of S then there exists a continuous and convergent automorphism

Since any rational power series (resp. polynomial), verifies the growth condition and its Lie rank is less or equal to its Hankel rank which is finite [START_REF] Fliess | Réalisation locale des systèmes non linéaires, algèbres de Lie filtrées transitives et séries génératrices non commutatives[END_REF] then Corollary 22. Any rational power series and any polynomial over X with coefficients in K are differentially produced.

Remark 5.

1. By Corollary 20, if S is polynomial then for any j = 1, .., d, S j is polynomial. Therefore, for i = 0, 1 and j = 1, .., d, x i ⊳ S is also polynomial over X. In this case, let (A, f ) be a differential representation of S of dimension d. Then f and {A i j } i=0,1 j=1,..,d are obviously polynomial on q1 , . . . , qd and the Lie algebra generated by {A(x i )} i=0,1 is nilpotent.

2. Note also that, by Theorem 6, if S is rational over X of linear representation (λ, µ, η) then the observation f (q 1 , . . . , q n ) equals λ 1 q 1 + . . . + λ n q n and the polysystem {A(x)} x∈X is obtained by putting

(µ(x i )) i j ∂ ∂q j yields linear representation not necessarily of minimal dimension [START_REF] Fliess | Réalisation locale des systèmes non linéaires, algèbres de Lie filtrées transitives et séries génératrices non commutatives[END_REF]. 35 In [START_REF] Fliess | Réalisation locale des systèmes non linéaires, algèbres de Lie filtrées transitives et séries génératrices non commutatives[END_REF][START_REF] Reutenauer | The local realisation of generating series of finite Lie rank[END_REF], the reader can found the discussion on the minimal differential representation.