Cal Remy

Pascal Remy
email: pascal.remy@uvsq.fr

Results in Mathematics Summability of inhomogeneous PDEs with a polynomial semilinearity Pas

Keywords: Summability, Inhomogeneous partial differential equation, Nonlinear partial differential equation, Formal power series, Divergent power series

In this article, we investigate the summability of the formal power series solutions in time of a certain class of inhomogeneous partial differential equations with a polynomial semilinearity, and with variable coefficients. In particular, we give necessary and sufficient conditions for the k-summability of the solutions in a given direction, where k is a positive rational number entirely determined by the linear part of the equation. These conditions generalize the ones given by the author for the linear case [?,?] and for the semilinear heat equation [?]. In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proof our main theorem.

Declarations

Not applicable 2 Introduction

Setting the problem

The summation theory is a very powerful tool initially developed within the framework of the analytic ordinary differential equations with an irregular singular point (see for instance [?, ?]). In particular, it allows the construction of explicit solutions from formal solutions.

For several years, various works have been done on the divergent solutions of some classes of linear partial differential equations or integro-differential equations in two variables or more, allowing thus to formulate many results on Gevrey properties, summability or multisummability (e.g. [?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?] and references inside).

In the case of the nonlinear partial differential equations, the situation is much more complicated. The existing results concern mainly Gevrey properties, especially the convergence (e.g. [?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?] and references inside), and there are very few results about the summation (see [?, ?, ?, ?, ?, ?]).

In this article, we are interested in the summability of the formal power series solutions in time of the inhomogeneous semilinear partial differential equation (2.1)

#

B κ t u ´apt, xqB p x u ´P puq " r f pt, xq B j t upt, xq |t"0 " φ j pxq, j " 0, ..., κ ´1

in two variables pt, xq P C 2 , where

• κ, p ě 1 are two positive integers;

• the coefficient apt, xq is analytic on a polydisc D ρ0 ˆDρ1 centered at the origin p0, 0q of C 2 (D ρ denotes the disc with center 0 P C and radius ρ ą 0) and satisfies the condition ap0, xq ı 0;

• P pXq " d ÿ m"2
b m pt, xqX m P OpD ρ0 ˆDρ1 qrXs is a polynomial in X with analytic coefficients on D ρ0 ˆDρ1 ;

• the inhomogeneity r f pt, xq is a formal power series in t with analytic coefficients in D ρ1 (we denote by r f pt, xq P OpD ρ1 qrrtss) which may be smooth, or not1 ;

• the initial conditions φ j pxq are analytic on D ρ1 for all j " 0, ..., κ ´1. Equation (2.1) is fundamental in many physical, chemical, biological, and ecological problems. For example: for pκ, pq " p1, 2q, Eq. (2.1) arises in problems involving diffusion and nonlinear growth such as heat and mass transfer, combustion theory, and spread theory of animal or plant populations (nonlinear heat equation); for pκ, pq " p2, 2q, Eq. (2.1) describes the propagation of nonlinear waves in an inhomogeneous medium (nonlinear Klein-Gordon equation); and, for pκ, pq " p2, 4q, Eq. (2.1) describes the relationship between the beam's deflection and an applied lateral nonlinear force (nonlinear Bernoulli-Euler equation).

The work presented in this article is a natural extension of the work [?] in which the nonlinearity P puq is reduced to a term of the form bpxqu 2 and pκ, pq " p1, 2q (see Eq. (2.2) just below). Before stating our main result (see Theorem 3) making explicit a characterization of the summability of the formal series solutions in time of Eq. (2.1), let us first start by recalling some known results about these ones.

Formal solutions and known results

First of all, we have the following.

Equation (2.1) admits a unique solution r upt, xq P OpD ρ1 qrrtss.

Let us write the coefficients apt, xq and b m pt, xq for m " 2, ..., d, and the inhomogeneity

r f pt, xq in the form apt, xq " ÿ jě0 a j,˚p xq t j j! , b m pt, xq " ÿ jě0 b m;j,˚p xq t j j! , r f pt, xq " ÿ jě0 f j,˚p xq t j j!
with a j,˚p xq, b m;j,˚p xq, f j,˚p xq P OpD ρ1 q for all j ě 0 and all m " 2, ..., d.

Looking for r upt, xq on the same type:

r upt, xq " ÿ jě0
u j,˚p xq t j j! with u j,˚p xq P OpD ρ1 q for all j ě 0, one easily checks that its coefficients u j,˚p xq are uniquely determined for all j ě 0 by the recurrence relations

u j`κ,˚p xq " f j,˚p xq `j ÿ j0"0 ˆj j 0 ˙aj0,˚p xqB p x u j´j0,˚p xqd ÿ m"2
ÿ j0`j1`...`jm"j ˆj j 0 , j 1 , ..., j m ˙bm;j0,˚p xqu j1,˚p xq...u jm,˚p xq, together with the initial conditions u j,˚p xq " φ j pxq for j " 0, ..., κ ´1. The notations ˆj j 0 ˙and ˆj j 0 , j 1 , ..., j m ˙stand respectively for the binomial coefficients and for the multinomial coefficients.

In a 1999 article [?], M. Miyake proved in the particular case of the equation

B κ t u ´Bp x u " 0
that the formal solution r upt, xq is convergent when κ ě p and s-Gevrey with s " p{κ ´1 otherwise. This result was then generalized by the author, first of all, to the inhomogeneous linear case P " 0 [?] and, afterwards, to the general Eq. (2.1) [?, ?]. In particular, he showed that the Gevrey regularity of r upt, xq does not depend on the nonlinear term P puq, but only on κ, p and r f pt, xq. [[?,?]] Let s be the nonnegative rational number defined by s " maxp0, p{κ1 q. Then, the formal solution r upt, xq and the inhomogeneity r f pt, xq are together s-Gevrey.

Thereby, in the case κ ě p, Proposition 2.2 provides us a necessary and sufficient condition for r upt, xq to be convergent, and, in the opposite case p ą κ, it naturally leads us to the question of the k-summability (k " 1{sq of r upt, xq. In the linear case, M. Miyake [?] and the author [?] gave necessary and sufficient conditions for r upt, xq to be k-summable in a given direction argptq " θ. To do that, they used two different approaches: the first one based on the definition of the k-summability in terms of the Borel transformation, and the second one based on the definition of the k-summability in terms of the successive derivatives.

More recently [?], the author considered Eq. (2.1) with pκ, p, dq " p1, 2, 2q and with constant coefficients in t, that is the semilinear heat equation (2.2)

# B t u ´apxqB 2 x u ´bpxqu 2 " r f pt, xq up0, xq " φpxq
Using the same approach as the one developed in [?], he gave a necessary and sufficient condition for r upt, xq to be 1-summable, generalizing thus the condition already proved in the linear case by W. Balser and M. Loday-Richaud [?]. In particular, he showed, as for the Gevrey regularity (see Proposition 2.2), that this condition is not affected by the nonlinear term u 2 .

In this article, we propose to extend all these results to the general Eq. (2.1). In Section 3, we recall some basic definitions and properties about the k-summable formal series. Then, we state our main result (Theorem 3) which gives, under some various assumptions on the coefficient apt, xq, a necessary and sufficient condition for r upt, xq to be k-summable in a given direction argptq " θ with k " 1{s " κ{pp ´κq. The proof of this result is developed in the next two sections. In Section 4, it is detailed in the case of the first assumption, namely ap0, 0q ‰ 0. Our approach is similar to the one presented in [?] for the linear case. However, because of the variable coefficients and the nonlinear terms u m , the calculations are much more complicated and require some technical results on the generalized binomial and multinomial coefficients, that is on the binomial and multinomial coefficients with nonnegative real terms. These technical results are all proved in Section 6, which can also be read independently of the rest of the article, so as not to burden the main proof. In Section 5, we show how to adapt the calculations of Section 4 within the framework of the other assumptions on apt, xq.

k-summability of r upt, xq

All along the article, we consider t as the variable and x as a parameter. Thereby, to define the notion of summability of formal power series in OpD ρ1 qrrtss, one extends the classical notion of summability of elements in Crrtss to families parametrized by x in requiring similar conditions, the estimates being however uniform with respect to x. Doing that, any formal power series r upt, xq P OpD ρ1 qrrtss can be seen as a formal power series in t with coefficients in a convenient Banach space defined as the space of functions that are holomorphic on a disc D r1 (0 ă r 1 ă ρ 1 ) and continuous up to its boundary, equipped with the usual supremum norm. For a general study of the series with coefficients in a Banach space, we refer for instance to [?].

Among the many equivalent definitions of the k-summability in a given direction argptq " θ at t " 0, we choose in this article a generalization of Ramis' definition which states that a formal series r gptq P Crrtss is k-summable in direction θ if there exists a holomorphic function g which is 1{k-Gevrey asymptotic to r

g in an open sector Σ θ,ąπs bisected by θ and with opening larger than πs with s " 1{k [?, Def. 3.1]. To express the 1{k-Gevrey asymptotic, there also exist various equivalent ways. We choose here the one which sets conditions on the successive derivatives of g (see [?, p. 171] or [?, Thm. 2.4] for instance).

[k-summability] Let k ą 0 and s " 1{k. A formal series r upt, xq P OpD ρ1 qrrtss is said to be k-summable in the direction argptq " θ if there exist a sector Σ θ,ąπs , a radius 0 ă r 1 ď ρ 1 and a function upt, xq called k-sum of r upt, xq in direction θ such that 1. u is defined and holomorphic on Σ θ,ąπs ˆDr1 ; 2. For any x P D r1 , the map t Þ Ñ upt, xq has r upt, xq "

ÿ jě0 u j,˚p xq t j j!
as Taylor series at 0 on Σ θ,ąπs ;

3. For any proper2 subsector Σ Ť Σ θ,ąπs , there exist two positive constants C ą 0 and K ą 0 such that, for all ℓ ě 0, all t P Σ and all x P D r1 , ˇˇB ℓ t upt, xq ˇˇď CK ℓ Γp1 `ps `1qℓq.

We denote by OpD ρ1 qttu k;θ the subset of OpD ρ1 qrrtss made of all the k-summable formal series in the direction argptq " θ.

Note that, for any fixed x P D r1 , the k-summability of r upt, xq coincides with the classical k-summability. Consequently, Watson's lemma [?, Theorem 5.1.3] implies the unicity of its k-sum, if any exists.

Note also that the k-sum of a k-summable formal series r upt, xq P OpD ρ1 qttu k;θ may be analytic with respect to x on a disc D r1 smaller than the common disc D ρ1 of analyticity of the coefficients u j,˚p xq of r upt, xq. Denote by B ´1 t r u (resp. B ´1 x r u) the anti-derivative of r u with respect to t (resp. x) which vanishes at t " 0 (resp. x " 0). Proposition 3 below specifies the algebraic structure of OpD ρ1 qttu k;θ .

Let k ą 0 and θ P R{2πZ. Then, pOpD ρ1 qttu k;θ , B t , B x q is a C-differential algebra stable under the anti-derivations B ´1 t and B ´1

x . We refer for instance to [?, Prop. 2] for a proof of this result.

With respect to t, the k-sum upt, xq of a k-summable series r upt, xq P OpD ρ1 qttu k;θ is analytic on an open sector for which there is no control on the angular opening except that it must be larger than π{k (hence, it contains a closed sector Σ θ,π{k bisected by θ and with opening π{k) and no control on the radius except that it must be positive. Thereby, the k-sum upt, xq is well-defined as a section of the sheaf of analytic functions in pt, xq on a germ of closed sector of opening π{k (that is, a closed interval I θ,π{k of length π{k on the circle S 1 of directions issuing from 0; see [?, 1.1] or [?, I.2]) times t0u (in the plane C of the variable x). We denote by O I θ,π{k ˆt0u the space of such sections.

The operator of k-summation

S k;θ : OpD ρ1 qttu k;θ ÝÑ O I θ,π{k ˆt0u r upt, xq Þ ÝÑ upt, xq
is a homomorphism of differential C-algebras for the derivations B t and B x . Moreover, it commutes with the anti-derivations B ´1 t and B ´1

x . We are now able to state the main result in this article. Let argptq " θ P R{2πZ be a direction issuing from 0. Let us assume p ą κ and let us set k " κ{pp ´κq. Let us also assume that either ap0, 0q ‰ 0, or there exists q P t1, ..., p ´1u such that B n x apt, xq |x"0 " 0 for all n " 0, ..., q ´1, and B q x ap0, 0q ‰ 0. Then, 1. The unique formal series solution r upt, xq P OpD ρ1 qrrtss of Eq. (2.1) is ksummable in the direction θ if and only if the inhomogeneity r f pt, xq and the formal series B n x r upt, xq |x"0 P Crrtss for n " 0, ..., p ´1 are k-summable in the direction θ.

2. Moreover, the k-sum upt, xq, if any exists, satisfies Eq. (2.1) in which r f pt, xq is replaced by its k-sum f pt, xq in the direction θ.

Observe that the necessary condition of the first point is straightforward from Proposition 3 and that the second point stems obvious from Corollary 3. Consequently, we are left to prove the sufficient condition of the first point. This is the subject of the next two sections below. In the first one (Section 4), we focus on the case ap0, 0q ‰ 0. In the second one (Section 5), we show how the calculations made in Section 4 can be adapted within the framework of the other assumptions on apt, xq.

From now on, we fix a direction θ and we suppose that the inhomogeneity r f pt, xq and the formal power series B n x r upt, xq |x"0 P Crrtss for n " 0, ..., p ´1 are all k-summable in the direction θ. To prove that the formal solution r upt, xq is also k-summable in this direction, we shall proceed through a fixed point method similar to the ones already used by W. Balser and M. Loday-Richaud in [?] and by the author in [?, ?, ?, ?]. However, as we shall see below, the calculations are much more complicated because of the nonlinear terms u m .

4 Proof of the sufficient condition: the case ap0, 0q ‰ 0

All along this section, we assume that the coefficient apt, xq satisfies ap0, 0q ‰ 0. Before starting the calculations, let us first begin this proof with a preliminary remark on the coefficients r u ˚,n ptq " B n x r upt, xq |x"0 of r upt, xq.

r upt, xq " ÿ ně0 r u ˚,n ptq x n n! , r f pt, xq " ÿ ně0 r f ˚,n ptq x n n! .
Observe that the coefficients r u ˚,n ptq and r f ˚,n ptq are divergent in general (hence, the notation with the tilde). By identifying the terms in x n in Eq. (2.1), we get the identities

(4.1) a ˚,0 ptqr u ˚,n`p ptq `n ÿ n0"1 ˆn n 0 ˙a˚,n0 ptqr u ˚,n`p´n0 ptq " B κ t r u ˚,n ptq ´r f ˚,n ptq ´d ÿ m"2 ÿ n0`n1`...`nm"n ˆn n 0 , n 1 , ..., n m ˙bm;˚,n0 ptqr u ˚,n1 ptq...r u ˚,nm ptq
for all n ě 0. By assumption, a ˚,0 p0q ‰ 0; hence, 1{a ˚,0 ptq is well-defined in Crrtss and, consequently, each coefficient r u ˚,n ptq is uniquely determined from the inhomogeneity r f pt, xq and from the formal series r u ˚,n 1 ptq with n 1 " 0, ..., p ´1. In particular, the same applies to r upt, xq. Observe that, thanks to the assumption ap0, 0q ‰ 0, these functions are all well-defined and holomorphic on a common convenient polydisc D ρ 1 0 ˆDρ 1 1 with 0 ă ρ 1 0 ď ρ 0 and 0 ă ρ 1 1 ď ρ 1 . According to our assumption on the k-summability of the inhomogeneity r f pt, xq and of the formal power series r u ˚,n ptq for n " 0, ..., p ´1, the formal series r vpt, xq and r gpt, xq are both k-summable in the direction θ (see Proposition 3). Thereby, the identity (4.2) above tells us that it is sufficient to prove that it is the same for the formal series r wpt, xq P OpD ρ1 qrrtss. To do that, we shall proceed as in [?, ?, ?, ?, ?] by using a fixed point method. Of course, as we shall see below, our calculations will be much more complicated due to the presence of the nonlinear terms B m pt, xqr v m´j pt, xqpB ´p x r wq j . recursively determined for all µ ě 0 by the relations

Second step: the associated equation

(4.3) r w µ`1 pt, xq " Apt, xqB κ t B ´p x r w µ pt, xq ´d ÿ m"2 m ÿ j"1 ÿ µ1`...`µj "µ « ˆm j ˙Bm pt, xqr v m´j pt, xq ˜j ź i 1 "1 B ´p x r w µ i 1 pt, xq
¸ff together with the initial condition r w 0 " r g. Observe that r w µ pt, xq P OpD ρ 1 1 qrrtss for all µ ě 0. Observe also that the r w µ pt, xq's are of order Opx pµ q in x for all µ ě 0, and, consequently, the series r wpt, xq itself makes sense as a formal series in t and x.

Let us now respectively denote by w 0 pt, xq and vpt, xq the k-sums of r w 0 and r v in the direction θ and, for all µ ą 0, let w µ pt, xq be determined by the relations (4.3) in which r v is replaced by v and all the r w µ are replaced by w µ . By construction, all the functions w µ pt, xq are defined and holomorphic on a common domain Σ θ,ąπs ˆDρ 2 1 , where s " 1{k " p{κ ´1, and where the radius ρ 2 0 of Σ θ,ąπs and the radius ρ 2 1 of D ρ 2 1 can always be chosen so that 0 ă ρ 2 0 ă ρ 1 0 and 0 ă ρ 2 1 ă ρ 1 1 . To end the proof, it remains to prove that the series ÿ µě0 w µ pt, xq is convergent and that its sum wpt, xq is the k-sum of r wpt, xq in the direction θ.

4.4 Fourth step: some estimates on the w µ pt, xq's

According to Definition 3, the k-summability of r w 0 and r v implies that there exists 0 ă r 1 1 ă minp1, ρ 2 1 q such that, for any proper subsector Σ Ť Σ θ,ąπs , there exist two positive constants C, K ą 0 such that, for all ℓ ě 0 and all pt, xq P Σ ˆDr 1 1 , the functions w 0 and v satisfy the inequalities (4.4) ˇˇB ℓ t w 0 pt, xq ˇˇď CK ℓ Γp1`ps`1qℓq and ˇˇB ℓ t vpt, xq ˇˇď CK ℓ Γp1`ps`1qℓq.

Let us now fix a proper subsector Σ Ť Σ θ,ąπs . Let r 1 0 denote the radius of Σ and let us choose for the constant K of the property above a constant

ě max ˆ1, 1 ρ 2 0 ´r1 0 ˙.
Observe that such a choice is already possible since the inequalities (4.4) still hold for any constant K 1 ě K. Observe also that the quotient 1{pρ 2 0 ´r1 0 q makes sense since the definition of a proper subsector (see Footnote 2) implies 0 ă r 1 0 ă ρ 2 0 . Proposition 4.4 below provides us some estimates on the derivatives B ℓ t w µ . Let us denote by

• α (resp. β m for m " 2, ..., d) the maximum of |Apt, xq| (resp. |B m pt, xq|)
on the closed polydisc D ρ 2 0 ˆDρ 2 1 (D ρ denotes the closed disc with center 0 P C and radius ρ ą 0);

• C s " s 1 p2 `Γpss 1 qq, where s 1 is the positive integer ě 1 defined by

s 1 " $ & % 1 if s ě 1 Z 1 s ^`1 if s ă 1
ptxu stands for the floor of x P Rq; hold for all ℓ, µ ě 0 and all pt, xq P Σ ˆDr 1 1 . Observe that the constant ζppq is well-defined since p ě 2. The proof proceeds by recursion on µ. The case µ " 0 is straightforward from the first inequality of (4.4). Let us now suppose that the inequalities (4.5) hold for all the functions w j pt, xq with j " 0, ..., µ for a certain µ ě 0.

• C 1 " ˜αC s `d ÿ m"2 m ÿ j"1 ˆm j ˙βm C m´1 C m s ζppq j´1 ¸,
According to the relations (4.3), we first derive from the generalized Leibniz Formula the identities

B ℓ t w µ`1 pt, xq " ℓ ÿ ℓ0"0 ˆℓ ℓ 0 ˙Bℓ´ℓ0 t Apt, xqB κ`ℓ0 t B ´p x w µ pt, xq ´d ÿ m"2 m ÿ j"1 ˆm j ˙« ÿ µ1`...`µj "µ ÿ ℓ0`ℓ1`...`ℓm"ℓ ˆℓ ℓ 0 , ℓ 1 , ..., ℓ m ˙Bℓ0 t B m pt, xqm ´j ź i"1 B ℓi t vpt, xq ¸˜j ź i 1 "1 B ℓ m´j`i 1 t B ´p x w µ i 1 pt, xq
¸ff for all ℓ ě 0 and all pt, xq P Σ ˆDr 1 1 , with the classical convention that the first product is 1 when j " m.

Let us now apply the Cauchy Integral Formula to the function B j t A with j ě 0. Thanks to the definition of the radii r 1 0 and r 1 1 , we have

B j t Apt, xq " j! p2iπq 2 ż |t 1 ´t|"ρ 2 0 ´r1 0 |x 1 ´x|"ρ 2 1 ´r1 1 Apt 1 , x 1 q pt 1 ´tq j`1 px 1 ´xq dt 1 dx 1 ,
for all pt, xq P Σ ˆDr 1 1 , and so the estimates

ˇˇB j t Apt, xq ˇˇď j!α ˆ1 ρ 2 0 ´r1 0 ˙j ď j!αK j
by definition of the constant K. In the same way, we have

ˇˇB j t B m pt, xq ˇˇď j!β m K j
for all j ě 0, all m " 2, ..., d and all pt, xq P Σ ˆDr 1 1 . Thereby, using the second inequality of (4.4), the inequalities (4.5) applied to the functions w j for j " 0, ..., µ, and the fact that r 1 1 ă 1 and K ě 1, we finally get the inequalities for all ℓ ě 0 and all pt, xq P Σ ˆDr 1 1 , with, as the constants S µ,ℓ and S 1 µ,ℓ,m,j , the constants respectively defined by S µ,ℓ " ℓ ÿ ℓ0"0 ˆℓ ℓ 0 ˙pℓ ´ℓ0 q!Γp1 `ps `1qpκµ `κ `ℓ0 qq Γp1 `ps `1qpκµ `κ `ℓqq and

S 1 µ,ℓ,m,j " ÿ µ1`...`µj "µ ÿ ℓ0`ℓ1`...`ℓm"ℓ ¨ˆℓ ℓ 0 , ℓ 1 , ..., ℓ m ˙ppµ `pq! j ź i 1 "1 ppµ i 1 `pq! l0 ! ˜m´j ź i"1 Γp1 `ps `1qℓ i q ¸˜j ź i 1 "1 Γp1 `ps `1qpκµ i 1 `ℓm´j`i 1 qq Γp1 `ps `1qpκµ `κ `ℓqq ‹ ‹ ‹ ‹ '
, where, as previously, the product on i is 1 when j " m.

The inequality (4.5) for w µ`1 stems then from Lemmas 4.4 and 4.4 below which allow to bound S µ,ℓ and S 1 µ,ℓ,m,j . This completes the proof of Proposition 4.4.

Let µ, ℓ ě 0 be. Then, S µ,ℓ ď C s . First of all, let us observe that pℓ ´ℓ0 q! ď Γp1 `ps `1qpℓ ´ℓ0 qq for all ℓ 0 P t0, ..., ℓu. This is obvious when ℓ 0 " ℓ and stems from the increasing of the Gamma function on r2, `8r otherwise. Thereby, using the notation of the generalized binomial coefficients (see Section 6) and the fact that ps `1qκ " p, we get

S µ,ℓ ď ℓ ÿ ℓ0"0 ˆℓ ℓ 0 ṗs `1qℓ `ppµ `1q ps `1qℓ 0 `ppµ `1q ˙.
Applying then the Vandermonde's inequality (see Proposition 6, 1) ˆps `1qℓ `ppµ `1q ps `1qℓ 0 `ppµ `1q Let µ, ℓ ě 0 be, m P t2, ..., du and j P t1, ..., mu. Then, S 1 µ,ℓ,m,j ď C m s ζppq j´1 . ‹ Let us start with the case j " 1. We must prove the inequality S 1 µ,ℓ,m,1 ď C m s , where

S 1 µ,ℓ,m,1 " ÿ ℓ0`ℓ1`...`ℓm"ℓ ˆℓ ℓ 0 , ℓ 1 , ..., ℓ m ˙l 0 ! ˜m´1 ź i"1
Γp1 `ps `1qℓ i q ¸Γp1 `ps `1qpκµ `ℓm qq Γp1 `ps `1qpκµ `κ `ℓqq .

Using the inequalities ℓ 0 ! ď Γp1 `ps `1qℓ 0 q and Γp1 `ps `1qpκµ `ℓm qq ď Γp1 `ps `1qpκpµ `1q `ℓm qq, and the fact that ps `1qκ " p, the sum S 1 µ,ℓ,m,1 can be first majorized in a similar way as the proof of Lemma 4.4 by means of generalized multinomial coefficients (see Section 6):

S 1 µ,ℓ,
ź i 1 "1 ppµ i 1 `pq! " 1 j´1 ź i 1 "1 ppµ i 1 `1q...ppµ i 1 `pq ˆpppµ `1qq! ppµ 1 q!...ppµ j´1 q!pppµ j `1qq! ď 1 j´1 ź i 1 "1 pµ i 1 `1q p ˆppµ `1q pµ 1 , ..., pµ j´1 , ppµ j `1q " 1 j´1 ź i 1 "1 pµ i 1 `1q p
ˆppµ `1q 0, ..., 0, pµ 1 , ..., pµ j´1 , ppµ j `1q

˙,

where 0 occurs m ´j times. Thereby, reasoning as in the case j " 1, we get

S 1 µ,ℓ,m,j ď ÿ µ1`...`µj "µ ¨1 j´1 ź i 1 "1 pµ i 1 `1q p ÿ ℓ0`ℓ1`...`ℓm"ℓ ˆℓ ℓ 0 , ℓ 1 , ..., ℓ m ˙ˆppµ `1q 0, ..., 0, pµ 1 , ..., pµ j´1 , ppµ j `1q Ṁµ,ℓ,m,j ‹ ‹ ' ,
where M µ,ℓ,m,j stands for the generalized multinomial coefficient ¨ps `1qℓ `ppµ `1q ps `1qℓ 0 , ps `1qℓ 1 , ..., ps `1qℓ m´j , ps `1qℓ m´j`1 `pµ 1 , ..., ps `1qℓ m´1 `pµ j´1 , ps `1qℓ m `ppµ j `1q ', Proposition 4.4 stems then from the inequality (4.5).

We are now able to complete the proof of Theorem 3.

Fifth step: conclusion

Let us now choose for Σ a sector containing a proper subsector Σ 1 bisected by the direction θ and opening larger than πs " π{k (such a choice is already possible by definition of a proper subsector, see Footnote 2). Let us also choose a radius 0 ă r 1 ă minpr 1 1 , c ´1{p q and let us set

C 1 :" C ÿ µě0 pcr p 1 q µ P R ˚.
Thanks to Proposition 4.4, the series ÿ µě0 B ℓ t w µ pt, xq are normally convergent on Σ ˆDr1 for all ℓ ě 0 and satisfy the inequalities

ÿ µě0 ˇˇB ℓ t w µ pt, xq ˇˇď C 1 K ℓ 1 Γp1 `ps `1qℓq
for all pt, xq P Σ ˆDr1 . In particular, the sum wpt, xq of the series

ÿ µě0 w µ pt, xq
is well-defined, holomorphic on Σ ˆDr1 and satisfies the inequalities

ˇˇB ℓ t wpt, xq ˇˇď C 1 K ℓ 1 Γp1 `ps `1qℓq
for all ℓ ě 0 and all pt, xq P Σ ˆDr1 . Hence, Conditions 1 and 3 of Definition 3 hold, since Σ 1 Ť Σ.

To prove the second condition of Definition 3, we proceed as follows. The removable singularities theorem implies the existence of lim This achieves the proof of the k-summability of r wpt, xq and, thereby, the fact that the condition is sufficient when ap0, 0q ‰ 0.

Proof of the sufficient condition: the other cases

In this section, we assume that there exists q P t1, ..., p´1u such that B n x apt, xq |x"0 " 0 for all n " 0, ..., q ´1, and B q x ap0, 0q ‰ 0.

Let us first observe that, under this assumption, the identities (4.1) become

(5.1) B κ t r u ˚,n ptq ´r f ˚,n ptq ´d ÿ m"2 ÿ n0`n1`...`nm"n ˆn n 0 , n 1 , ..., n m ˙bm;˚,n0 ptqr u ˚,n1 ptq...r u ˚,nm ptq " 0 for all n ď q ´1, and

(5.2) ˆn q ˙a˚,q ptqr u ˚,n`p´q ptq `n ÿ n0"q`1 ˆn n 0 ˙a˚,n0 ptqr u ˚,n`p´n0 ptq " B κ t r u ˚,n ptq ´r f ˚,n ptq ´d ÿ m"2 ÿ n0`n1`...`nm"n ˆn n 0 , n 1 , ..., n m ˙bm;˚,n0 ptqr u ˚,n1 ptq...r u ˚,nm ptq for all n ě q, with a ˚,q p0q ‰ 0. In particular, the identities (5.2) tell us, as in the case ap0, 0q ‰ 0, that each coefficient r u ˚,n ptq (hence, the formal solution r upt, xq too) is uniquely determined from the inhomogeneity r f pt, xq and from the formal series r u ˚,n 1 ptq with n 1 " 0, ..., p ´1. Observe also that our assumption allows us to write the function apt, xq in the form apt, xq " x q a q pt, xq with a q p0, 0q ‰ 0. Thereby, the functions A q pt, xq " 1 a q pt, xq and B m;q pt, xq " b m pt, xq a q pt, xq with m " 2, ..., d, are all well-defined and holomorphic on a convenient common polydisc centered at the origin p0, 0q P C 2 , say D ρ 1 0 ˆDρ 1 1 with 0 ă ρ 1 0 ď ρ 0 and 0 ă ρ 1 1 ď ρ 1 to use the same notations as the case ap0, 0q ‰ 0. Setting as before r upt, xq " r vpt, xq `B´p x r wpt, xq with

r vpt, xq " p´1 ÿ n"0 r u ˚,n ptq x n n! ,
Eq. (2.1) becomes now (5.3)

r w ´Aq pt, xq x q B κ t B ´p x r w `d ÿ m"2 m ÿ j"1 ˆm j ˙Bm;q pt, xq x q r v m´j pt, xqpB ´p x r wq j " r gpt, xq where r gpt, xq " A q pt, xq B κ t r vpt, xq ´d ÿ m"2 b m pt, xqr v m pt, xq ´r f pt, xq
x q is again a formal power series in t and x. Indeed, due to the identities (5.1), the term

B κ t r vpt, xq ´d ÿ m"2 b m pt, xqr v m pt, xq ´r f pt, xq
is of order Opx q q in x. Assuming then r vpt, xq and r gpt, xq to be k-summable in the direction θ, we can prove as in the case ap0, 0q ‰ 0 that r wpt, xq is also k-summable in the direction θ. The proof being similar to the one developed in Section 4, we give below only the key points to modify. In particular, we keep all the notations on the choices of the sectors Σ and Σ 1 , and on the choices of the various radii.

First of all, let us start by observing that the r w µ pt, xq's are now recursively determined for all µ ě 0 by the relations

(5.4) r w µ`1 pt, xq " A q pt, xq x q B κ t B ´p x r w µ pt, xq ´d ÿ m"2 m ÿ j"1 ÿ µ1`...`µj "µ « ˆm j ˙Bm;q pt, xq x q r v m´j pt, xq ˜j ź i 1 "1 B ´p x r w µ i 1 pt, xq
¸ff together with the initial condition r w 0 " r g. In particular, the operator 1 x q B ´p x in place of B ´p x implies that the r w µ pt, xq's are of order Opx pp´qqµ q in x for all µ ě 0, instead of Opx pµ q as in the case ap0, 0q ‰ 0. Still, r wpt, xq is again a formal power series in t and in x since p ´q ě 1

In doing so, the estimates on the derivatives B ℓ t w µ given in Proposition 4.4 are modified as follows.

For all ℓ, µ ě 0 and all pt, xq P Σ ˆDr 1 1 , ˇˇB ℓ t w µ pt, xq ˇˇď CC 1µ K κµ`ℓ Γp1 `ps `1qpκµ `ℓqq |x| pp´qqµ pµ!q q ppp ´qqµq! with

C 1 " ˜αq C s `d ÿ m"2 m ÿ j"1 ˆm j ˙βm;q C m´1 C m s ζppq j´1 ¸,
where α q (resp. β m;q for m " 2, ..., d) stands for the maximum of |A q pt, xq| (resp. |B m;q pt, xq|) on the closed polydisc D ρ 2 0 ˆDρ 2 1 . We proceed as in the proof of Proposition 4.4 by replacing the inequality (4.6) by the inequality (5.5)

ˇˇB ℓ t w µ`1 pt, xq ˇˇď CC 1µ K κpµ`1q`ℓ Γp1 `ps `1qpκpµ `1q `ℓqq |x| pp´qqpµ`1q
ppµ `1q!q q ppp ´qqpµ `1qq! αq

S µ,ℓ,q `d ÿ m"2 m ÿ j"1 ˆm j ˙βm;q C m´1 S 1 µ,ℓ,m,j,q ¸,
where the constants S µ,ℓ,q and S 1 µ,ℓ,m,j,q are respectively defined by S µ,ℓ,q " ℓ ÿ ℓ0"0 ˆℓ ℓ 0 ˙pℓ ´ℓ0 q!Γp1 `ps `1qpκµ `κ `ℓ0 qq Γp1 `ps `1qpκµ `κ `ℓqq F µ,ℓ,q and S 1 µ,ℓ,m,j,q " ÿ µ1`...`µj "µ ÿ ℓ0`ℓ1`...`ℓm"ℓ

ˆˆℓ ℓ 0 , ℓ 1 , ..., ℓ m ˙F 1 µ,ℓ,j,q l0 ! ˜m´j ź i"1 Γp1 `ps `1qℓ i q ¸˜j ź i 1 "1 Γp1 `ps `1qpκµ i 1 `ℓm´j`i 1 qq Γp1 `ps `1qpκµ `κ `ℓqq ‹ ‹ ‹ ‹ '
, with F µ,ℓ,q " ppµ `1q!q q ppp ´qqpµ `1qq! pµ!q q ppp ´qqµq!ppp ´qqµ `1q...ppp ´qqµ `pq and F 1 µ,ℓ,j,q " ppµ `1q!q q ppp ´qqpµ `1qq! j ź i 1 "1 pµ i 1 !q q ppp ´qqµ i 1 q!ppp ´qqµ i 1 `1q...ppp ´qqµ i 1 `pq .

As previously, the product on i in S 1 µ,ℓ,m,j,q is 1 when j " m. Observe that F 1 µ,ℓ,1,q " F µ,ℓ,q . Observe also that the term F µ,ℓ,q is obvious ď 1. Indeed, the inequalities p ą p ´q ě 1 imply F µ,ℓ,q " ppµ `1q!q q pµ!q q ppp ´qqµ `p ´q `1q...ppp ´qqµ `pq ď ppµ `1q!q q pµ!q q pµ `1q q " 1.

Thereby, Lemmas 4.4 and 4.4 provide us the inequalities S µ,ℓ,q ď S µ,ℓ ď C s and S 1 µ,ℓ,m,1,q ď S 1 µ,ℓ,m,1 ď C m s " C m s ζppq 0 .

By reasoning in the same way for F 1 µ,ℓ,j,q when j ě 2, we first have F 1 µ,ℓ,j,q ď ppµ `1q!q q ppp ´qqpµ `1qq! where µ occurs q times in the multinomial coefficient. Hence, the following Let us set K 1 " 2 p K and c " p2q `2q p C 1 K κ . Then, the inequalities

j ź i 1 "1 ppµ i 1 `1q!q q ppp ´qqpµ i 1 `1qq! " ˆµ `1 µ 1 , ...,
ˇˇB ℓ t w µ pt, xq ˇˇď CK ℓ 1 Γp1 `ps `1qℓq ´c |x| p´q ¯µ
hold for all ℓ, µ ě 0 and all pt, xq P Σ ˆDr 1 1 . The end of the proof is similar to the one of the case ap0, 0q ‰ 0 and is left to the reader. This completes the proof of Theorem 3.

6 Some technical results on the generalized binomial and multinomial coefficients

In combinatorial analysis, the binomial coefficients ˆn m ˙and the multinomial coefficients ˆn n 1 , ..., n q ˙are defined for any nonnegative integers 0 ď m ď n and any tuples pn, n 1 , ..., n q q of nonnegative integers satisfying q ě 2 and n 1 `... ǹq " n by the relations

ˆn m ˙" n! k!pn ´kq! and ˆn n 1 , ..., n q ˙" n! n 1 !...n q ! .
They respectively denote the number of ways of choosing m objects from a collection of n distinct objects without regard to order, and the number of ways of putting n " n 1 `... `nq different objects into q different boxes with n i in the i-th box for all i " 1, ..., q.

Using the fact that n! " Γp1`nq for any integer n ě 0, one can easily extend the definitions of these coefficients to the case where their terms are no longer necessarily integers by setting (6.1) ˆa b ˙" Γp1 `aq Γp1 `bqΓp1 `a ´bq for any nonnegative real numbers 0 ď b ď a and (6.2) ˆa a 1 , ..., a q ˙" Γp1 `aq Γp1 `a1 q...Γp1 `aq q " Γp1 `aq

q ź i"1
Γp1 `ai q for any tuples pa, a 1 , ..., a q q of nonnegative real numbers satisfying q ě 2 and a 1 `... `aq " a. Observe that all these coefficients are positive. Observe also that one has the following decomposition (6.3) ˆa a 1 , ..., a q ˙" q ź

i"2 ˆa1 `... `ai a 1 `... `ai´1

˙.

The four propositions below extend to the generalized binomial coefficients (6.1) and the generalized multinomial coefficients (6.2) some well-known results in combinatorial analysis.

In the proof of Theorem 3 (see Sections 4 and 5), we essentially used the inequalities stated in Propositions 6 and 6. The result of Proposition 6 is useful for the proof of Proposition 6 and the one of Proposition 6 is used in the proof of Proposition 6.

[Pascal's formula] Let 0 ď b ď a be two nonnegative real numbers and 1 ď m ď n two nonnegative integers. Then, (6.4)

ˆa `n `1 b `m ˙" ˆa `n b `m˙`ˆa `n b `m ´1˙.
We compute:

ˆa `n b `m˙`ˆa `n b `m ´1˙" Γp1 `a `nq Γp1 `b `mqΓp1 `a ´b `n ´mq `Γp1 `a `nq Γp1 `b `m ´1qΓp1 `a ´b `n ´m `1q " pa ´b `n ´m `1qΓp1 `a `nq Γp1 `b `mqΓp1 `a ´b `n ´m `1q `pb `mqΓp1 `a `nq Γp1 `b `mqΓp1 `a ´b `n ´m `1q " pa `n `1qΓp1 `a `nq Γp1 `b `mqΓp1 `a ´b `n ´m `1q " Γp1 `a `n `1q Γp1 `b `mqΓp1 `a ´b `n ´m `1q " ˆa `n `1 b `m ˙;
hence, the identity (6.4). 2. (Multinomial case) Let q ě 2 be an integer, pa, a 1 , ..., a q q a tuple of nonnegative real numbers and pn, n 1 , ..., n q q a tuple of nonnegative integers such that a 1 `... `aq " a and n 1 `... `nq " n. Then, (6.6) ˆa `n a 1 `n1 , ..., a q `nq ˙ě ˆa a 1 , ..., a q ˙ˆn n 1 , ..., n q ˙.

‹ First point. The inequality (6.5) is clear for n " m " 0. Let us now fix 0 ď b ď a and let us prove by induction on n ě 1 the property pP n q : @m P t0, ..., nu, Assuming now the property pP n q for a certain n ě 1, let us prove the property pP n`1 q. As for the property pP 1 q, the sought inequality stems from a direct calculation when m " 0 and m " n `1: When m P t1, ..., nu, it stems from Proposition 6 and the property pP n q as follows: This ends the induction and proves the first point of Proposition 6.

ˆa `n `1 b `m ˙" ˆa
‹ Second point. Let us apply the relation (6.3) and the inequality (6.5) to each factor of the product. We get ˆa `n a 1 `n1 , ..., a q `nq ˙" q ź

i"2 ˆa1 `. The inequality (6.6) follows then by applying again the relation (6.3), which ends the proof of the second point of Proposition 6. Let s, a ą 0 be. Then, the function In particular, B a pbq ě 1 for all b P r0, as.

The derivative of B a is defined for all b P r0, as by B 1 a pbq " sB a pbqpΨp1 `spa ´bqq ´Ψp1 `sbqq, where Ψ " Γ 1 {Γ is the Psi (or Digamma) function. The latter being increasing on r0, as (the function ln Γ is indeed convex on s0, `8r), Proposition 6 follows from Lagrange Theorem.

[Sum of the inverses of binomial and multinomial coefficients] Let s ą 0 be and let us set C s " s 1 p2 `Γpss 1 qq, where s 1 is the positive integer ě 1 defined by

s 1 " $ & % 1 if s ě 1 Z 1 s ^`1 if s ă 1 ,
where txu stands for the floor of x P R.

(Binomial case)

The following inequality holds for all integers n ě 0:

(6.7) n ÿ m"0 1 ˆsn sm ˙ď C s .

(Multinomial case)

The following inequality holds for all integers q ě 2 and n ě 0:

(6.8) ÿ n1`...`nq"n 1 ˆsn sn 1 , ..., sn q ˙ď C q´1 s .

4. 1

 1 First step: a preliminary remark Let us write the coefficients apt, xq and b m pt, xq for m " 2, ..., d in the form apt, xq " ÿ ně0 a ˚,n ptq x n n! , b m pt, xq " ÿ ně0 b m;˚,n ptq x n n! with a ˚,n ptq, b m;˚,n ptq P OpD ρ0 q for all n ě 0 and all m " 2, ..., d. Let us also write the formal solution r upt, xq and the inhomogeneity r f pt, xq in the same way:

4. 3

 3 Third step: the fixed point procedure Let us set r wpt, xq " ÿ µě0 r w µ pt, xq and let us choose the solution of Eq. (4.2)

  pt, xq ˇˇď CC 1µ K κµ`ℓ Γp1 `ps `1qpκµ `ℓqq |x| pµ ppµq!

  w µ`1 pt, xq ˇˇď CC 1µ K κpµ`1q`ℓ Γp1 `ps `1qpκpµ `1q

  case) Let 0 ď b ď a be two nonnegative real numbers and 0 ď m ď n two nonnegative integers. Then,

B

  a : b P r0, as Þ ÝÑ

  it is clear that wpt, xq satisfies equation (4.2) with vpt, xq in place of r vpt, xq and right-hand side gpt, xq in place of r gpt, xq and, consequently, so does its Taylor series. Then, since equation (4.2) has a unique formal series solution r wpt, xq, we then conclude that the Taylor expansion of wpt, xq is r wpt, xq. Hence, Condition 2 of Definition 3 holds.

	tPΣ 1 tÑ0	B ℓ t wpt, xq for all

x P D r1 and, thereby, the existence of the Taylor series of w at 0 on Σ 1 for all x P D r1 (see for instance [?,

Cor. 1.1.3.3]

; see also

[?, Prop. 1.1.11]

). On the other hand, considering recurrence relations (4.3) with w µ pt, xq and the ksums vpt, xq and gpt, xq instead of r w µ pt, xq, r vpt, xq and r gpt, xq,

  µ j´1 , µ j `1˙ˆp p ´qqpµ `1q pp ´qqµ 1 , ..., pp ´qqµ j´1 , pp ´qqpµ j `1q pt, xq ˇˇď Cp2 p Kq ℓ p2 p C 1 K κ |x|

	Applying then the calculations made in the proof of Lemma 4.4, we finally get
	S 1 µ,ℓ,m,j,q ď	ÿ	¨1 j´1		ÿ	1 ˆsℓ	˙‹ ‹ ‹ ‹
			µ1`...`µj "µ	ź i 1 "1 pµ i 1 `1q p	ℓ0`ℓ1`...`ℓm"ℓ	sℓ 0 , sℓ 1 , ..., sℓ m	'
		ď C m s ζppq j´1 ,
	which ends the proof of Proposition 5.
	Using now the inequality
		Γp1 `ps `1qpκµ `ℓqq ď 2 pℓ`pµ ppµq!Γp1 `ps `1qℓq
	proved in the proof of Proposition 4.4, Proposition 5 leads us to the inequality
	ˇˇB ℓ t w µ
							j´1
			ź	pµ i 1 `1q q ppp ´qqµ i 1 `1q...ppp ´qqµ i 1 `p ´qq
			i 1 "1
	ď	ˆµ µ 1 , ..., µ j´1 , µ j `1	`1˙ˆp p ´qqpµ `1q pp ´qqµ 1 , ..., pp ´qqµ j´1 , pp ´qqpµ j `1q	j´1 .
					ź	pµ i 1 `1q p
					i 1 "1

p´q q µ Γp1 `ps `1qℓq ˆˆpµ µ, ..., µ, pp ´qqµ ˙,

  .. `ai `n1 `... `ni a 1 `... `ai´1 `n1 `... `ni´1

						ě
	q ź	ˆa1 `... `ai	˙ˆn 1 `... `ni	"
	i"2		a 1 `... `ai´1		n 1 `... `ni´1
	˜q ź	ˆa1 `... `ai	˙¸˜q ź	ˆn1 `... `ni	˙¸.
	i"2	a 1 `... `ai´1	i"2	n 1 `... `ni´1

We denote r f with a tilde to emphasize the possible divergence of the series r f .

A subsector Σ of a sector Σ 1 is said to be a proper subsector and one denotes Σ Ť Σ 1 if its closure in C is contained in Σ 1 Y t0u.

This completes the proof.

The following result, which provides the estimates on the w µ 's in view in this section, is a direct consequence of Proposition 4.4.

Let us set K 1 " 2 p K and c " 2 p C 1 K κ . Then, the inequalities

µ hold for all ℓ, µ ě 0 and all pt, xq P Σ ˆDr 1 1 . Using the relation ps `1qκ " p, we first derive from the recurrence relation Γp1 `zq " zΓpzq applied pµ times the identity Γp1 `ps `1qpκµ `ℓqq " Γp1 `ps `1qℓq

Next, applying the inequality s `1 ď p, we get the following relations:

ď 2 pℓ`pµ ppµq!Γp1 `ps `1qℓq.

‹ First point. The inequality (6.7) is straightforward from Proposition 6 when n ă 2s 1 since we have the relations

Let us now assume n ě 2s 1 and let us write the left hand-side of (6.7) in the form

Applying Proposition 6, the first and the third sums of the right hand-side are both ď s 1 . Moreover, all the terms of the second sum are ě ˆsn ss 1 ˙. This brings us to the following relations:

Γp1 `sn ´ss 1 q Γp1 `sn ´1q ď 2s 1 `s1 Γpss 1 q Γp1 `sn ´ss 1 q Γp1 `sn ´1q .

The inequality (6.7) follows then from the increasing of the Gamma function on r2, `8r. Indeed, the inequality ss 1 ě 1 implies 2 ď 1 `sn ´ss 1 ď 1 `sn ´1, and thereby Γp1 `sn ´ss 1 q Γp1 `sn ´1q ď 1.

‹ Second point. Applying the relation (6.3) and setting n 1 k " n 1 `... `nk for all k " 1, ..., q ´1, we first get the identities The inequality (6.8) stems then from the inequality (6.7) which we apply q ´1 times. This completes the proof.