Mohammed Mesk

Mohammed Brahim Zahaf

ON A CLASS OF POLYNOMIALS GENERATED BY F (xt -R(t))

Keywords: Generating functions, d-orthogonal polynomials, recurrence relations, generalized hypergeometric series AMS Subject Classification: 12E10, 33C47, 33C20

We investigate polynomial sets {P n } n≥0 with generating power series of the form F (xt -R(t)) and satisfying, for n ≥ 0, the (d + 1)-order recursion xP n (x) = P n+1 (x) + d l=0 γ l n P n-l (x), where {γ l n } is a complex sequence for 0 ≤ l ≤ d, P 0 (x) = 1 and P n (x) = 0 for all negative integer n. We show that the formal power series R(t) is a polynomial of degree at most d + 1 if certain coefficients of R(t) are null or if F (t) is a generalized hypergeometric series. Moreover, for the d-symmetric case we demonstrate that R(t) is the monomial of degree d + 1 and F (t) is expressed by hypergeometric series.

Introduction

In [START_REF] Al-Salam | On a characterization of a certain set of orthogonal polynomials[END_REF][START_REF] Von Bachhaus | The orthogonal polynomials generated by F (xt -αt 2 ) = n≥0 α n P n (x)t n[END_REF][START_REF] Cheikh | On the Classical d-Orthogonal Polynomials Defined by Certain Generating Functions, 1[END_REF] the authors used different methods to show that the orthogonal polynomials defined by a generating function of the form F (xt -αt2 ) are the ultraspherical and Hermite polynomials. On the other hand, the author in [START_REF] Anshelevich | A characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind[END_REF] found (even if F is a formal power series) that the orthogonal polynomials are the ultraspherical, Hermite and Chebychev polynomials of the first kind. Motivated by the problem, posed in [START_REF] Anshelevich | A characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind[END_REF], of describing (all or just orthogonal) polynomials with generating functions F (xU (t) -R(t)) we have generalized in [START_REF] Mesk | A new characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind[END_REF] the above results by proving the following:

Theorem 1 [START_REF] Mesk | A new characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind[END_REF] Let F (t) = n≥0 α n t n and R(t) = n≥1 R n t n /n be formal power series where {α n } and {R n } are complex sequences with α 0 = 1 and R 1 = 0. Define the polynomial set {P n } n≥0 by

F (xt -R(t)) = n≥0 α n P n (x)t n . ( 1
)
If this polynomial set (which is automatically monic) satisfies the three-term recursion relation xP n (x) = P n+1 (x) + β n P n (x) + ω n P n-1 (x), n ≥ 0, P -1 (x) = 0, P 0 (x) = 1 [START_REF] Anshelevich | A characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind[END_REF] where {β n } and {ω n } are complex sequences, then we have: a) If R 2 = 0 and α n = 0 for n ≥ 1, then R(t) = 0, F (t) is arbitrary and F (xt) = n≥0 α n x n t n generates the monomials {x n } n≥0 . b) If α 1 R 2 = 0, then R(t) = R 2 t 2 /2 and the polynomial sets {P n } n≥0 are the rescaled ultraspherical, Hermite and Chebychev polynomials of the first kind.

Note that, the polynomials in Theorem 1 which satisfy a three term recursion with complex coefficients are not necessary orthogonal with respect to a moment functional L, i.e. for all non negative integers m, n; L(P m (x)P n (x)) = 0 if m = n and L(P 2 n (x)) = 0, see Definition 2.2 in [START_REF] Chihara | An introduction to orthogonal polynomials[END_REF].

Remark 1 a) A polynomial set PS, {P n } n≥0 , is such that degree(P n ) = n, n ≥ 0. b) A PS is called a monic PS if

P n (x) = x n + • • • , for n ≥ 0.
c) The choice α 0 = 1 and R 1 = 0 comes from the fact that the generating function γ 1 + γ 2 F ((x + R 1 )t -R(t)) = γ 1 + γ 2 n≥0 α n P n (x + R 1 )t n , with γ 1 and γ 2 constants, is also of type [START_REF] Al-Salam | On a characterization of a certain set of orthogonal polynomials[END_REF].

In the present paper, we are interested in monic PSs generated by (1) (with F (t) and R(t) as in Theorem 1) and satisfying higher order recurrence relations [START_REF] Von Bachhaus | The orthogonal polynomials generated by F (xt -αt 2 ) = n≥0 α n P n (x)t n[END_REF]. For this purpose, we adopt the following definitions:

Definition 1 Let d ∈ N. A PS {Q n } n≥0 is called a d-polynomial set d-PS if its corresponding monic PS {P n } n≥0 , defined by P n (x) = (lim x→+∞ x -n Q n (x)) -1 Q n (x)
, n ≥ 0, satisfies the (d + 1)-order recurrence relation:

xP n (x) = P n+1 (x) + d l=0 γ l n P n-l (x), n ≥ 0, P 0 (x) = 1, P -l (x) = 0, 1 ≤ l ≤ d

where {γ l n } n≥0 , 0 ≤ l ≤ d, are complex sequences (4) and {γ d n } n≥d is not the null sequence, for d ≥ 1.

(

) 5
Definition 2 Let {P n } n≥0 be a d-PS. If the PS of the derivatives {(n + 1) -1 P ′ n+1 } n≥0 is also a d-PS, then {P n } n≥0 is called a classical d-PS.

Definition 3 [START_REF] Douak | Les polynômes orthogonaux "classiques" de dimension deux[END_REF] Let ω = exp(2iπ/(d + 1)), where i 2 = -1. The PS {P n } n≥0 is called d-symmetric if it fulfils:

P n (ωx) = ω n P n (x), n ≥ 0.

Remark 2 In Definition 1: a) For d ≥ 1, the first terms {γ l n } 0≤n<l≤d of the sequences {γ l n } n≥0 , 1 ≤ l ≤ d, are arbitrary. b) For d = 0, (3) becomes xP n (x) = P n+1 (x) + γ 0 n P n (x), n ≥ 0, with P 0 (x) = 1. Here {γ 0 n } n≥0 can be the null sequence, so the set of monomials is a 0-PS.

An interesting class of d-PSs characterized by [START_REF] Von Bachhaus | The orthogonal polynomials generated by F (xt -αt 2 ) = n≥0 α n P n (x)t n[END_REF], with the additional condition γ d n = 0 for n ≥ d, are the d-orthogonal polynomial sets d-OPSs [START_REF] Maroni | L'orthogonalité et les récurrences de polynômes d'ordre supérieur á deux[END_REF][START_REF] Van Iseghem | Vector orthogonal relations. Vector QD-algorithm[END_REF]. In this context, the authors in [START_REF] Cheikh | On the Classical d-Orthogonal Polynomials Defined by Certain Generating Functions, 1[END_REF] generalized the result stated in [START_REF] Al-Salam | On a characterization of a certain set of orthogonal polynomials[END_REF][START_REF] Von Bachhaus | The orthogonal polynomials generated by F (xt -αt 2 ) = n≥0 α n P n (x)t n[END_REF] by showing the following:

Theorem 2 [START_REF] Cheikh | On the Classical d-Orthogonal Polynomials Defined by Certain Generating Functions, 1[END_REF] The only d-OPSs generated by G((d + 1)xt -t d+1 ) are the classical d-symmetric polynomials.

Another contribution concerns d-OPSs with generating functions of Sheffer type, i.e. of the form A(t) exp(xH(t)). We have Theorem 3 [START_REF] Varma | A characterization theorem and its applications for d-orthogonality of Sheffer polynomial sets[END_REF] Let ρ d (t) = d k=0 ρk t k be a polynomial of degree d (ρ d = 0) and σ d+1 (t) = d+1 k=0 σk t k be a polynomial of degree less than or equal to d + 1. The only PSs, which are d-orthogonal and also Sheffer PS, are generated by

exp t 0 ρ d (s) σ d+1 (s) ds exp x t 0 1 σ d+1 (s) ds = ∞ k=0 P n (x) t n n! (7)
with the conditions σ0 (nσ

0 -ρd ) = 0, n ≥ 1. (8)
Note that Theorem 3 characterizes also the d-OPSs with generating functions of the form

F (xH(t)-R(t)) with F (t) = exp(t), since A(t) exp(xH(t)) = F (xH(t) -R(t)) where R(t) = -ln(A(t))
. For H(t) = t we meet the Appell case with the Hermite d-OPSs [START_REF] Douak | The relation of the d-orthogonal polynomials to the Appell polynomials[END_REF] generated by exp(xt -ρd+1 (t)) where ρd+1 (t) is a polynomial of degree d + 1. As a consequence of the results obtained in this paper, we give some generalizations of Theorems 1, 2 and the Appell case in Theorem 3 (see also [START_REF] Douak | The relation of the d-orthogonal polynomials to the Appell polynomials[END_REF]) to d-PS generated by [START_REF] Al-Salam | On a characterization of a certain set of orthogonal polynomials[END_REF].

After this short introduction, we give in section 2 some results for d-PSs generated by [START_REF] Al-Salam | On a characterization of a certain set of orthogonal polynomials[END_REF]. Then in section 3 we show that the only d-symmetric d-PSs generated by [START_REF] Al-Salam | On a characterization of a certain set of orthogonal polynomials[END_REF] are the classical d-symmetric polynomials. For this later case, we give in section 4 the (d + 1)-order recurrence relation [START_REF] Von Bachhaus | The orthogonal polynomials generated by F (xt -αt 2 ) = n≥0 α n P n (x)t n[END_REF] and the expression of F (t) by means of hypergeometric functions.

Some general results

The results in this section concern all d-PSs generated by [START_REF] Al-Salam | On a characterization of a certain set of orthogonal polynomials[END_REF]. The central result is Proposition 2 below from which the other results arise. First we have Proposition 1 [START_REF] Mesk | A new characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind[END_REF] Let {P n } n≥0 be a PS generated by (1). Then we have

α n xP ′ n (x) - n-1 k=1 R k+1 α n-k P ′ n-k (x) = nα n P n (x), n ≥ 1. (9)
Secondly Proposition 2 Let {P n } be a d-PS generated by (1) and satisfying (3), ( 4) and (5), with α n = 0 for n ≥ 1. Putting

a n = α n α n+1 , (n ≥ 0) and c l n = α n α n-l γ l n , (1 ≤ l ≤ d, n ≥ l),
then we have: a)

γ 0 n = 0, for n ≥ 0. (10)
b)

γ 1 n = R 2 2 (na n -(n -1)a n-1 ), for n ≥ 1. ( 11
)
or equivalently

c 1 n = R 2 2 n a n a n-1 -(n -1) , for n ≥ 1. ( 12
)
c)

c 2 n = R 3 3 (n -1) a n a n-2 -(n -2) , for n ≥ 2. ( 13
)
d.i) k + 1 n -k + 1 a n-k c k n = R k+1 a n - n -k n -k + 1 a n-k + k-2 l=1 R k-l n + 2 n -l + 1 c l n - n -k + l + 1 n -k + l + 2 c l n-k+l+1 - k-2 l=1 l + 1 n -l + 1 c l n c k-l-1 n-l - k-2 l=1 R l+1 R k-l n -l + 1 , 3 ≤ k ≤ d, n ≥ k. ( 14
) d.ii) R k+1 a n - n -k n -k + 1 a n-k + d l=1 R k-l n + 2 n -l + 1 c l n - n -k + l + 1 n -k + l + 2 c l n-k+l+1 - d l=k-1-d l + 1 n -l + 1 c l n c k-l-1 n-l = k-2 l=1 R l+1 R k-l n -l + 1 , d + 1 ≤ k ≤ 2d + 1, n ≥ k. ( 15
) d.iii) R k+1 a n - n -k n -k + 1 a n-k + d l=1 R k-l n + 2 n -l + 1 c l n - n -k + l + 1 n -k + l + 2 c l n-k+l+1 = k-2 l=1 R l+1 R k-l n -l + 1 , k ≥ 2d + 2, n ≥ k. ( 16
)
Proof: By differentiating (3) we get

xP ′ n (x) + P n (x) = P ′ n+1 (x) + d l=0 γ l n P ′ n-l (x). ( 17
)
Then by making the operations nα n Eq(17) + Eq [START_REF] Douak | Les polynômes orthogonaux "classiques" de dimension deux[END_REF] and Eq(9) -α n Eq(17) we obtain, respectively,

(n + 1)α n xP ′ n (x) = nα n P ′ n+1 (x) + d l=0 γ l n P ′ n-l (x) + n-1 k=1 R k+1 α n-k P ′ n-k (x) (18)
and

(n + 1)α n P n (x) = α n P ′ n+1 (x) + d l=0 γ l n P ′ n-l (x) - n-1 k=1 R k+1 α n-k P ′ n-k (x). ( 19
)
Inserting [START_REF] Douak | Les polynômes orthogonaux "classiques" de dimension deux[END_REF] in the left-hand side of the equation Eq(19) multiplied by x we obtain

(n + 1)α n P n+1 + d l=0 γ l n P n-l (x) = α n xP ′ n+1 (x) + d l=0 γ l n xP ′ n-l (x) - n-1 k=1 R k+1 α n-k xP ′ n-k (x). (20)
Using ( 19) and ( 18) respectively in the left hand side and right hand side of (20) we get

n + 1 n + 2 α n P ′ n+2 (x) + d l=0 γ l n+1 P ′ n+1-l (x) - n + 1 n + 2 α n α n+1 n k=1 R k+1 α n+1-k P ′ n+1-k (x) +(n + 1)α n d l=0 γ l n 1 n + 1 -l P ′ n+1-l (x) + (n + 1)α n d l=0 γ l n 1 n + 1 -l d l ′ =0 γ l ′ n-l P ′ n-l-l ′ (x) -(n + 1)α n d l=0 γ l n 1 n + 1 -l 1 α n-l n-l-1 k=1 R k+1 α n-l-k P ′ n-k-l (x) = n + 1 n + 2 α n P ′ n+2 (x) + d l=0 γ l n+1 P ′ n+1-l (x) + 1 n + 2 α n α n+1 n k=1 R k+1 α n+1-k P ′ n+1-k (x) + α n d l=0 γ l n n -l n + 1 -l P ′ n+1-l (x) +α n d l=0 γ l n n -l n + 1 -l d l ′ =0 γ l ′ n-l P ′ n-l-l ′ (x) + α n d l=0 γ l n 1 n + 1 -l 1 α n-l n-l-1 k=1 R k+1 α n-l-k P ′ n-l-k (x) - n-1 k=1 R k+1 n -k n + 1 -k α n-k P ′ n+1-k (x) - n-1 k=1 R k+1 n -k n + 1 -k α n-k d l=0 γ l n-k P ′ n-k-l (x) - n-1 k=1 R k+1 1 n + 1 -k n-k-1 k ′ =1 R k ′ +1 α n-k-k ′ P ′ n-k-k ′ (x). ( 21
)
It follows that

- α n α n+1 n k=1 R k+1 α n+1-k P ′ n+1-k (x) + α n d l=0 γ l n l + 1 n + 1 -l P ′ n+1-l (x) + α n d l=0 γ l n l + 1 n + 1 -l d l ′ =0 γ l ′ n-l P ′ n-l-l ′ (x) -(n + 2)α n d l=0 γ l n 1 n + 1 -l 1 α n-l n-l-1 k=1 R k+1 α n-l-k P ′ n-k-l (x) = - n-1 k=1 R k+1 n -k n + 1 -k α n-k P ′ n+1-k (x) - n-1 k=1 R k+1 n -k n + 1 -k α n-k d l=0 γ l n-k P ′ n-k-l (x) - n-1 k=1 R k+1 1 n + 1 -k n-k-1 k ′ =1 R k ′ +1 α n-k-k ′ P ′ n-k-k ′ (x) (22)
a) By comparing the coefficients of P ′ n+1 (x) in the both sides of (22) we obtain

1 n + 1 α n γ 0 n = 0, for n ≥ 0,
and then γ 0 n = 0, for n ≥ 0. b) Equating the coefficients of P ′ n (x) in the both sides of the equation ( 22) gives

2 n α n γ 1 n = R 2 α n α n+1 α n -R 2 n -1 n α n-1 , for n ≥ 1,
which can be written as γ 1 n = na n -(n -1)a n-1 , for n ≥ 1. Now by equating the coefficients of P ′ n+1-k (x) for k ≥ 2 in the both sides of the equation ( 22) we obtain

k + 1 n + 1 -k α n γ k n + R k+1 α n-k n -k n + 1 -k -R k+1 α n α n+1 α n+1-k + α n d l=1 γ l n l + 1 n + 1 -l d l ′ =1 γ l ′ n-l δ k-1 l+l ′ -(n + 2)α n d l=1 γ l n 1 n + 1 -l n-l-1 k ′ =1 R k ′ +1 α n-l-k ′ α n-l δ k-1 l+k ′ + n k ′ =1 R k ′ +1 α n-k ′ n -k ′ n + 1 -k ′ d l=1 γ l n-k ′ δ k-1 k ′ +l + n k ′′ =1 R k ′′ +1 1 n + 1 -k ′′ n-k ′′ k ′ =1 R k ′ +1 α n-k ′ -k ′′ δ k-1 k ′ +k ′′ = 0, for n ≥ k. ( 23
)
then by taking k = 2 in (23) we retrieve c) and by considering 3 ≤ k ≤ d, d + 1 ≤ k ≤ 2d + 1 and k ≥ 2d + 2 we obtain d.i.), d.ii.) and d.iii.) respectively.

In the following corollaries we adopt the same conditions and notations as in Proposition 2.

Corollary 1 If R 2 = R 3 = • • • = R d+1 = 0 and α n = 0 for n ≥ 1, then R(t) = 0, F (t) is arbitrary and F (xt) = n≥0 α n x n t n generates the monomials {x n } n≥0 . Proof: As R 1 = R 2 = • • • = R d+1 = 0, it is enough to show by induction that R n = 0 for n ≥ d + 2.
For n = 1, 2, ..., d + 2, the equation ( 9) gives P 1 (x) = x, P 2 (x) = x 2 , ..., P d+1 (x) = x d+1 and P d+2 (0) = -R d+2 α 1 (d+2)α d+2 . But according to equation (3), for n = d + 1, P d+2 (0) = 0 and then R d+2 = 0. Now assume that R k = 0 for d + 2 ≤ k ≤ n -1. According to [START_REF] Douak | Les polynômes orthogonaux "classiques" de dimension deux[END_REF] we have, for d + 2 ≤ k ≤ n -1, P k (0) = 0 and P n (0) = -Rnα 1 nαn . On other hand, by the shift n → n -1 in (3) we have P n (0) = 0 and thus R n = 0. As R(t) = 0, the generating function (1) reduces to F (xt) = n≥0 α n x n t n which generates the monomials with F (t) arbitrary.

Corollary 2 If R d+2 = R d+3 = • • • = R 2d+2 = 0 then R(t) = R 2 t 2 /2 + R 3 t 3 /3 + • • • + R d+1 t d+1 /(d + 1).
Proof: We will use [START_REF] Varma | A characterization theorem and its applications for d-orthogonality of Sheffer polynomial sets[END_REF] and proceed by induction on k to show that R k = 0 for k ≥ 2d + 3. Indeed k = 2d + 2 and n = 2d + 2 in ( 16) leads to a 2d+2 R 2d+3 = 0 and since a n = 0 we get 16) gives a k R k+1 = 0 and finally R k+1 = 0.

R 2d+3 = 0. Suppose that R 2d+3 = R 2d+4 = • • • = R k = 0, then for n = k the equation (
Corollary 3 If R κ+d+1 = • • • = R κ+1 = R κ = R κ-1 = • • • = R κ-d = 0 for some κ ≥ 3d + 3, then R d+2 = R d+3 = • • • = R 2d+2 = 0. Proof: • Let k = κ in (16), then for n ≥ κ the fraction κ-2 l=1 R l+1 R κ-l n-l+1 , as function of integer n, is null even for real n. So, lim x→l-1 (x -l + 1) κ-2 s=1 R s+1 R κ-s x -s + 1 = R l+1 R κ-l = 0, for 1 ≤ l ≤ κ -2 (24) which is R d+2 R κ-d-1 = 0 when l = d+1. Supposing R d+2 = 0 leads to R κ-d-1 = 0. So R κ+d = R κ+d-1 = • • • = R κ-d = R κ-d-1
= 0 and with the same procedure we find R κ-d-2 = 0. Going so on till we arrive at R d+2 = 0 which contradicts R d+2 = 0.

• By taking successively k = κ + r, κ + r -1, ..., κ in ( 16), for 1 ≤ r ≤ d, we find

R l+1 R κ+r-l = 0 for 1 ≤ l ≤ κ + r -2, R l+1 R κ+r-1-l = 0 for 1 ≤ l ≤ κ + r -3, . . . R l+1 R κ-l = 0 for 1 ≤ l ≤ κ -2. If R d+2+r = 0 then by taking l = d + 1 + r we get R κ-d-1 = R κ-d-2 = • • • = R κ-d-r-1 = 0. So R κ+d-r = R κ+d-r-1 = • • • = R κ-d-r-1 = 0 and with the same procedure we find R κ-d-r-2 = R κ-d-r-3 = • • • = R κ-d-2r-2 = 0.
Going so on till we arrive at R d+2+r = 0 which contradicts R d+2+r = 0.

Corollary 4 If a n is a rational function of n then R d+2 = R d+3 = • • • = R 2d+2 = 0.
Proof: From ( 12), ( 13) and ( 14) observe that c l n will also be a rational function of n. Then it follows that, in [START_REF] Varma | A characterization theorem and its applications for d-orthogonality of Sheffer polynomial sets[END_REF], two fractions are equal for natural numbers n ≥ k, k ≥ 2d + 2, and consequently will be for real numbers n. If we denote by N s (G(x)) the number of singularities of a rational function G(x) then we can easily verify, for all rational functions G and G of x and a constant a = 0, that:

a) N s (G(x + a)) = N s (G(x)), b) N s (aG(x)) = N s (G(x)), c) N s (G(x) + G(x)) ≤ N s (G(x)) + N s ( G(x)).
Using property a) of N s we have

N s n -k n -k + 1 a n-k = N s n n + 1 a n and N s n -k + l + 1 n -k + l + 2 c l n-k+l+1 = N s n n + 1 c l n .
According to properties b) and c) of N s , the N s of the left-hand side of ( 15) is finite and independent of k. Thus, the right-hand side of ( 15) has a finite number of singularities which is independent of k. As consequence there exists a

k 1 ≥ 3d + 3 for which R l+1 R k-l = 0 for all k ≥ k 1 -d -1 and k 1 -d -1 ≤ l ≤ k. According to Corollary 1, there exists a k 0 such that 2 ≤ k 0 ≤ d + 1 and R k 0 = 0. So, taking successively k = k 0 + l with l = k 1 + d, k 1 + d -1, ..., k 1 -d -1 we get R k 1 +d+1 = R k 1 +d = • • • = R k 1 -d = 0. Then, by Corollary 3 we have R d+2 = R d+3 = • • • = R 2d+2 = 0.
The fact that a n is a rational function of n means that F (ǫz) = n≥0 α n (ǫz) n (where ǫ is the quotient of the leading coefficients of the numerator and the denominator of a n ) is a generalized hypergeometric series, i.e. of the form:

p F q (µ l ) p l=1 (ν l ) q l=1 ; z = p F q µ 1 , µ 2 , ..., µ p ν 1 , ν 2 , ..., ν q ; z = n≥0 (µ 1 ) n (µ 2 ) n • • • (µ p ) n (ν 1 ) n (ν 2 ) n • • • (ν q ) n z n n! (25)
where (µ l ) p l=k denotes the array of complex parameters µ k , µ k+1 , ..., µ p , and if k > p we take the convention that (µ l ) p l=k is the empty array. The symbol (µ) n stands for the shifted factorials, i.e.

(µ

) 0 = 1, (µ) n = µ(µ + 1) • • • (µ + n -1), n ≥ 1. ( 26
)
As an interesting consequence, from Corollary 4 and Corollary 2 we state the following result, which can be interpreted as a generalization of the Appell case in the above Theorem 3 (see also [START_REF] Douak | The relation of the d-orthogonal polynomials to the Appell polynomials[END_REF]):

Theorem 4 Let {P n } be a d-PS generated by (1) with F (z) a generalized hypergeometric series. Then R(t) = R 2 t 2 /2 + R 3 t 3 /3 + • • • + R d+1 t d+1 /(d + 1). Proof: F (z) = n≥0 α n z n has the form (25). Then a n = α n /α n+1 is a rational function of n, since (µ) n+1 /(µ) n = n + µ.
The use of Corollary 4 and Corollary 2 completes the proof.

Corollary 5 Let R(t) = R 2 t 2 /2 + R 3 t 3 /3 + • • • + R d+1 t d+1 /(d + 1). Then, i) If R d+1 = 0 we have c d n+d c d n = 0, for n ≥ d + 1. ii) If R d+1 = 0 then c d n = R d+1 d + 1 (n + 1) b n-d b n -(n -d) , for n ≥ d + 1, ( 27
)
where b md+r = (b d+r -b r )m + b r , f or m ≥ 0, 1 ≤ r ≤ d.
iii) The {c m n } 1≤m≤d-1 can be calculated recursively by solving the following d-order linear difference equations:

1 n -m + 1 R d+1 (n + 2) -(n -d)c d n-m c m n - 1 n -d + 1 R d+1 (n -d) + (d + 1)c d n c m n-d + + d l=m+1 R m+d+1-l n + 2 n -l + 1 c l n - n -m -d + l n -m -d + l + 1 c l n-m-d+l - d-1 l=m+1 l + 1 n -l + 1 c l n c m+d-l n-l = d-1 l=m R l+1 R m+d+1-l n -l + 1 , 1 ≤ m ≤ d -1, n ≥ m + d + 1. ( 28
)
Proof:

The proof of i) Put k = 2d + 1 in [START_REF] Van Iseghem | Vector orthogonal relations. Vector QD-algorithm[END_REF] to get the following Riccati equation for {c d n }:

R d+1 (n + 2) c d n -(n -d) c d n-d -(d + 1)c d n c d n-d -R 2 d+1 = 0, f or n ≥ 2d + 1. ( 29
)
By taking R d+1 = 0 in (29), i) follows immediately.

The proof of ii) Substituting ( 27) in ( 29) we find the 2d-linear homogeneous equation

b n -2b n-d + b n-2d = 0, f or n ≥ 2d + 1. ( 30
)
By writing n = md + r, where m, r are natural numbers with 1 ≤ r ≤ d, the equation (30) can be solved by summing twice to find that

b md+r = (b d+r -b r )m + b r , f or m ≥ 0, 1 ≤ r ≤ d.
The proof of iii)

Since d + 1 ≤ k ≤ 2d + 1 we have R k+1 = 0 and R k-l = 0 for l ≤ k -2 -d. So, we can write (15) as d l=k-d R k-l n + 2 n -l + 1 c l n - n -k + l + 1 n -k + l + 2 c l n-k+l+1 + R d+1 n + 2 n -k + d + 2 c k-1-d n - n -d n -d + 1 c k-1-d n-d - d-1 l=k-d l + 1 n -l + 1 c l n c k-l-1 n-l - k -d n -k + d + 2 c k-1-d n c d n-k+1+d - d + 1 n -d + 1 c d n c k-1-d n-d = = k-2 l=k-1-d R l+1 R k-l n -l + 1 , d + 1 ≤ k ≤ 2d + 1, n ≥ k. ( 31
)
Putting m = k -d -1 = 0 in (31) and rearranging we obtain (28).

Remark 3 In the case of d-OPSs, in Corollary 5 the polynomial R(t) is of degree d + 1. Otherwise (i.e. R d+1 = 0), we have a contradiction with the regularity conditions γ d n = 0, for n ≥ d.

Corollary 6 The d-PS is classical if and only if R(t) = R 2 t 2 /2 + R 3 t 3 /3 + • • • + R d+1 t d+1 /(d + 1) with R d+1 = 0.
Proof: 1) Assume that the d-PS is classical. From (18) and Definition 2 we have

R k+1 = 0 for d + 1 ≤ k ≤ n -1.
We get R(t) by taking n ≥ 2d + 2 and using Corollary 2. Now we show that R d+1 = 0. Equation ( 18) becomes

xQ n (x) = Q n+1 (x) + d l=1 γl n Q n-l (x), n ≥ 0, ( 32
)
where

Q n (x) = (n + 1) -1 P ′ n+1 (x) and γl n = n + 1 -l n + 2 γ l n+1 + R l+1 α n+l-1 (n + 1)α n+1 , n ≥ d. (33)
From ( 27), if R d+1 = 0 then γ d n+1 = 0, and (33) gives γd

n = 0, for n ≥ d. So, {Q n } is not a d-PS which contradicts the fact that {P n } is classical (see Definition 2). 2) Assume that R(t) = R 2 t 2 /2 + R 3 t 3 /3 + • • • + R d+1 t d+1 /(d + 1
) with R d+1 = 0 , then the PS of the derivatives {Q n } satisfy (32) and are generated by

F ′ (xt -R(t)) = n≥0 (n + 1)α n+1 Q n (x)t n . Using Corollary 5 we find that cd n := (n + 1)α n+1 γd n /((n -d + 1)α n-d+1
) satisfies (29). And according to the same expression (29), we should have, if c d n = 0 or γ d n = 0 (for n ≥ d + 1), R d+1 = 0. Therefore, there exists for cd n , since R d+1 = 0, a n 0 ≥ d + 1 such that cd n 0 = 0 or γd n 0 = 0. This means that {P n } is classical.

The d-symmetric case

The main result of this section is the following:

Theorem 5 If {P n } is a d-symmetric d-PS generated by (1) then R(t) = R d+1 t d+1 /(d + 1).
Theorem 5 generalizes Theorem 1 and Theorem 2 mentioned above. Its proof is quite similar to that of Theorem 1 in [START_REF] Mesk | A new characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind[END_REF] and it requires the following Lemmas.

Lemma 1 If {P n } is a d-symmetric d-PS generated by (1) then R(t) = k≥1 R k(d+1) k(d + 1) t k(d+1) . ( 34
)
Proof: Let {P n } be a d-symmetric d-PS satisfying (3) and generated by [START_REF] Al-Salam | On a characterization of a certain set of orthogonal polynomials[END_REF]. Then it has, according to Definition 3, the property

P n (ωx) = ω n P n (x), (35)
where ω = exp(2πi/(d + 1)). It follows that (3) becomes [START_REF] Douak | Les polynômes orthogonaux "classiques" de dimension deux[END_REF] xP n (x) = P n+1 (x) + γ d n P n-d (x), n ≥ 0, P -n (x) = 0, 1 ≤ n ≤ d, and P 0 (x) = 1.

(36)

Let us show that R k = 0 when k is not a multiple of d + 1. First we replace x by ωx in (9) and use (35

) with P ′ n (ωx) = ω n-1 P ′ n (x) to get α n xP ′ n (x) - n-1 k=1 R k+1 α n-k ω -k-1 P ′ n-k (x) = nα n P n (x), n ≥ 2. ( 37
)
Subtracting (37) from [START_REF] Douak | Les polynômes orthogonaux "classiques" de dimension deux[END_REF] gives

n-1 k=1 R k+1 α n-k (1 -ω -k-1 )P ′ n-k (x) = 0, n ≥ 2 (38) which leads to R k α n-k+1 (1 -ω -k ) = 0, for 2 ≤ k ≤ n, n ≥ 2.
Since ω k = 1, provided k is not a multiple of d + 1, gives the result.

By Lemma 1 and putting T k = R k(d+1) for k ≥ 0, the equations in Proposition 2 simplify to particular forms. Indeed, from ( 12), ( 13) and ( 14) we get

c d n = T 1 d + 1 (n -d + 1) a n a n-d -(n -d) , for n ≥ d. (39)
The equation [START_REF] Van Iseghem | Vector orthogonal relations. Vector QD-algorithm[END_REF], with k = 2d + 1, becomes

T 2 a n - n -2d -1 n -2d a n-2d-1 + T 1 n + 2 n -d + 1 c d n - n -d n -d + 1 c d n-d - d + 1 n -d + 1 c d n c d n-d = T 2 1 n -d + 1 , n ≥ 2d + 1, (40)
which by (39) takes the form

(d + 1)T 2 T 2 1 1 - n -2d -1 n -2d a n-2d-1 a n = n + 1 a n - 2(n -d + 1) a n-d + n -2d + 1 a n-2d , for n ≥ 2d + 1. (41)
Finally, the equation ( 16) simplifies to

T k+1 a n - n -k(d + 1) -d n -k(d + 1) -d + 1 a n-k(d+1)-d + T k n + 2 n -d + 1 c d n - n -k(d + 1) + 1 n -k(d + 1) + 2 c d n-k(d+1)+1 = k-1 l=0 T l+1 T k-l n -l(d + 1) -d + 1 , k ≥ 2, n ≥ k(d + 1) + d. ( 42
)
This equation will be denoted by E k (n) in below. Proof: The proof is similar to that of Corollary 7 in [START_REF] Mesk | A new characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind[END_REF]. Let assume that T κ+1 = 0 and T m+1 = 0, since if not, we apply Corollary 3. When m > κ and by using (42), the following operations

Lemma 2 If T 2 = 0 then R(t) = T 1 t d+1 /(d + 1). Proof: According to Corollary 2, if T 2 = R 2(d+1) = 0 then R(t) = T 1 t d+1 /(d + 1), since in this case we have R d+2 = R d+3 = • • • = R 2d+1 = 0. Lemma 3 If T m = T m+1 = 0 for some m ≥ 3, then T 2 = 0. Proof: T m = T m+1 = 0 means that R (d+1)m = R (d+1)(m+1) = 0. Also by Lemma 1, we have R (d+1)m+d = R (d+1)m+d-1 = • • • = R (d+1)m+1 = 0 and R (d+1)m-1 = R (d+1)m-2 = • • • = R (
[E κ (n + (d + 1)m + d)/T κ+1 -E m (n + (d + 1)m + d)/T m+1 ] -[E κ (n)/T κ+1 -E m (n + (d + 1)(m -κ) + d)/T m+1 ] (43)
give

n n + 1 - n + (d + 1)(m -κ) n + (d + 1)(m -κ) + 1 a n = Q(n) (44)
where Q(n) is a rational function of n. Consequently, a n is a rational function of n and by Corollary 4 we have T 2 = 0.

Lemma 5

The following equality is true for k ≥ 3 and n ≥ k(d + 1) + 2d + 1.

T k-1 D k+1 (a n -ãn-k(d+1)-2d-1 ) -T k+1 D k (a n-d-1 -ãn-k(d+1)-d ) = k-1 l=1 V k,l n -l(d + 1) -d + 1 , (45)
where

• D k,l = T k T k-l+1 -T k+1 T k-l . • D k = D k,1 = T 2 k -T k+1 T k-1 . • V k,l = T 1 2 (T l T k+1 D k-1,l-1 -T l+1 T k-1 D k,l ).
• ãn = n n+1 a n .

Proof: Just by making the following combinations it is easy to get (45):

T k+1 (T k-1 E k (n) -T k E k-1 (n -d -1)) -T k-1 (T k E k+1 (n) -T k+1 E k (n -d - 1
)) .

To prove Theorem 5 it is sufficient, according to Lemma 2, to show that T 2 = 0. To this end, we will consider three cases:

Case 1: There exists k 0 ≥ 3 such that D k = 0 for k ≥ k 0 . and 3a n+d+1 n + d + 2 -D 2a n n + 1 - a n-d-1 n -d = Q (4) k (n). ( 54
)
By shifting n → n + d + 1 in (53) we obtain

3a n-d-1 n -d -D - a n+d+1 n + d + 2 + 2a n n + 1 = Q (3) k (n + d + 1). ( 55
)
The coefficients a n+d+1 and a n-d-1 can be eliminated by the operations D × Eq(54) -3 Eq(55) and 3 Eq(54) -D × Eq(55) leaving us with

6D -2D 2 n + 1 a n + D 2 -9 n -d a n-d-1 = Q (5) k (n) (56) and 9 -D 2 n + d + 2 a n+d+1 - 6D -2D 2 n + 1 a n = Q (6) k (n). ( 57
)
Finally, the shifting n → n -d -1 in (57) leads to

9 -D 2 n + 1 a n - 6D -2D 2 n -d a n-d-1 = Q (6) k (n -d -1) (58)
and the operation (6D -2D 2 )Eq(56) + (D 2 -9)Eq(58) gives

[(6D -2D 2 ) 2 + (D 2 -9) 2 ]a n = Q (7) k (n). ( 59
)
According to manipulations made above, Q

k (n) is a rational function of n. As consequence, if D = 3, a n is a rational function of n and then T 2 = 0. Now, we explore the case D = 3. According to the left-hand sides of (54) and (55), we have

Q (3) k (n + d + 1) = Q (4) k (n) ,
which can be written as

(n + 2d + 2)Q (1)
k (n + d + 1) -(n -2d -2)Q (1) k (n) = (n + 2d + 3) Q (1) k (n + d + 1) -(n -2d -1) Q (1) k (n) .
(60) By using, from (46) and (48), the expressions of

Q (1) k (n) and Q (1) k (n) with W k,k+1 = W k,k+1 = W k,0 = W k,0 = 0 we obtain k l=0 ((d + 1)l + 2d)W k,l+1 -((d + 1)l -2d -4)W k,l n -(d + 1)l + 2 = k l=0 (d + 1)((2 -l) W k,l -(l + 2) W k,l+1 ) n + (d + 1)l + 1 . (61)
Observe that in (61) the singularities of the left hand side are different from those of the right hand side. So,

((d + 1)l + 2d)W k,l+1 -((d + 1)l -2d -4)W k,l = (2 -l) W k,l -(l + 2) W k,l+1 = 0, (0 ≤ l ≤ k), (62)
and by induction on l, all the W k,l and W k,l are null. Thus, (46) reads

a n+d+1 -a n-2(d+1) -3(a n -a n-d-1 ) = 0. ( 63
)
For n = (d + 1)m + r, m ≥ 0 and 0 ≤ r ≤ d, the solutions of (63) have the form

a (d+1)m+r = C 0,r + C 1,r m + C 2,r m 2 , ( 64
)
where C 0,r , C 1,r and C 2,r are constants. So, by Corollary 4 we get T 2 = 0.

Case 2: There exists

k 0 ≥ 3 such that D k = 0 for k ≥ k 0 . Suppose that D k = T 2 k -T k-1 T k+1 = 0 for all k ≥ k 0 . First, notice that if there exists a k 1 ≥ k 0 such that T k 1 = 0, then T k 1 -1 T k 1 +1 = 0. Then, T k 1 -1 = 0 or T k 1 +1
= 0 and by Corollary 3, T 2 = 0. We have also T k 0 -1 = 0, otherwise T k 0 = 0 and by Corollary 3, T 2 = 0. Now, for T k = 0 (k ≥ k 0 -1), we have

T k+1 T k = T k T k-1 = T k 0 T k 0 -1 . ( 65
)
This means that

T k = T k 0 T k 0 -1 k-k 0 T k 0 = ab k (66)
where

a = T k 0 k 0 -1 /T k 0 -1 k 0 = 0 and b = T k 0 /T k 0 -1 = 0. The substitution T k = ab k in (42) for k ≥ k 0 leads to the equation b a n - n -k(d + 1) -d n -k(d + 1) -d + 1 a n-k(d+1)-d + n + 2 n -d + 1 c d n - n -k(d + 1) + 1 n -k(d + 1) + 2 c d n-k(d+1)+1 = b -k a k-1 l=0 T l+1 T k-l n -l(d + 1) -d + 1 = Q k (n) . (67)
Let denote (67) by E (k, n) and make the subtraction

E (k + 1, n + d + 1) -E (k, n) to get b (a n+d+1 -a n ) + n + d + 3 n + 2 c d n+d+1 - n + 2 n -d + 1 c d n = Q k+1 (n + d + 1) -Q k (n) . ( 68
)
On the right hand side of (68) we have, for k ≥ k 0 , the expression

Q k (n) := Q k+1 (n + d + 1) -Q k (n) = b -k-1 a k l=0 T l+1 T k+1-l n + d + 1 -l(d + 1) -d + 1 - b -k a k-1 l=0 T l+1 T k-l n -l(d + 1) -d + 1 = b -k-1 a k l=0 T l+1 T k+1-l n -l(d + 1) + 2 - b -k a k l=1 T l T k+1-l n -l(d + 1) + 2 = b -k-1 a T k+1 T 1 n + 2 + b -k-1 a T k (T 2 -bT 1 ) n -d + 1 + b -k-1 a k l=2 T k+1-l (T l+1 -bT l ) n -l(d + 1) + 2 = T 1 n + 2 + T 2 -bT 1 b(n -d + 1) + b -k-1 a k l=2 T k+1-l (T l+1 -bT l ) n -l(d + 1) + 2 . ( 69
)
from which we deduce

Q k+1 (n) = T 1 n + 2 + T 2 -bT 1 b(n -d + 1) + b -k-2 a k+1 l=2 T k+2-l (T l+1 -bT l ) n -l(d + 1) + 2 . ( 70
)
Now since the left hand side of equation ( 68) is independent of k, it follows

Q k+1 (n) -Q k (n) = b -k-2 a k l=2 (T k-l+2 -bT k-l+1 ) (T l+1 -bT l ) n -l(d + 1) + 2 = 0. (71)
As a result, for 2 ≤ l ≤ k and k ≥ k 0 , we have

(T k-l+2 -bT k-l+1 ) (T l+1 -bT l ) = 0. ( 72
)
Let take k = 2 (k 0 -2) -1 and l = k 0 -2 to get (T k 0 -1 -bT k 0 -2 ) 2 = 0 and then T k 0 -1 = bT k 0 -2 , (or equivalently D k 0 -1 = 0). Thus, the equations ( 65) and (66) are valid for k = k 0 -1 and by induction we arrive at T 4 = bT 3 , (or equivalently D 4 = 0). For k = 4, the right-hand side of (45) is null. Consequently, V 4,2 = 0 and using T 5 = T 2 4 /T 3 (from D 4 = 0) we get D 3 = 0. On the other side (when T 2 = 0) we can write

T k = T 3 T 2 k-2 T 2 = ab k , for k ≥ 2,
where b = T 3 /T 2 = 0 and a = T 3 2 /T 2 3 = 0. Therefore, the equation (67) reads

b a n - n -k(d + 1) -d n -k(d + 1) -d + 1 a n-k(d+1)-d + n + 2 n -d + 1 c d n - n -k(d + 1) + 1 n -k(d + 1) + 2 c d n-k(d+1)+1 = T 1 n -(d + 1)k + 2 + T 1 n + k-2 l=1 ab n -l(d + 1) -d + 1 , k ≥ 2 and n ≥ (d + 1)k + d. (73)
When n = (d + 1)k + d and n = (d + 1)(k + 1), the equation (73) gives

ba (d+1)k+d + (d + 1)k + d + 2 (d + 1)k + 1 c d (d+1)k+d = d + 1 d + 2 c d d+1 + T 1 d + 2 + T 1 (d + 1)k + d + k-2 l=1 ab (d + 1)(k -l) + 1 (74) and ba (d+1)(k+1) + (d + 1)(k + 1) + 2 (d + 1)(k + 1) -d + 1 c d (d+1)(k+1) = a 1 b d + 1 + d + 2 d + 3 c d d+2 + T 1 d + 3 + T 1 (d + 1)(k + 1) + k-2 l=1 ab (d + 1)(k + 1 -l) -d + 1 (75)
respectively. Let take n = (d + 1)N + d in (73) and use (74) to obtain the expression

d + 1 d + 2 c d d+1 + T 1 d + 2 + T 1 (d + 1)N + d + N -2 l=1 ab (d + 1)(N -l) + 1 - (d + 1)(N -k)b (d + 1)(N -k) + 1 a (d+1)(N -k) - (d + 1)(N -k + 1) (d + 1)(N -k) + d + 2 c d (d+1)(N -k+1) = T 1 (d + 1)(N -k) + d + 2 + T 1 (d + 1)N + d + k-2 l=1 ab (d + 1)(N -l) + 1 .
In this last equality let put N -k instead of k to get

- b(d + 1)k (d + 1)k + 1 a (d+1)k - (d + 1)(k + 1) (d + 1)(k + 1) + 1 c d (d+1)(k+1) = = - (d + 1) d + 2 c d d+1 - T 1 d + 2 - N -2 l=1 ab (d + 1)(N -l) + 1 + T 1 (d + 1)k + d + 2 + N -k-2 l=1 ab (d + 1)(N -l) + 1 = - d + 1 d + 2 c d d+1 - T 1 d + 2 + T 1 (d + 1)k + d + 2 - k+1 l=2 ab (d + 1)l + 1 . ( 76
)
After defining

A 1 = a 1 d+1 + d+2 (d+3)b c d d+2 + T 1 (d+3)b , A 2 = -d+1 d+2 c d d+1 b -T 1 (d+2)b and A 3 = T 1 b , the operation 1 (d + 1)(k + 1) + 2 (d + 1)(k + 1) (d + 1)(k + 1) + 1 Eq(75) + (d + 1)(k + 1) + 2 (d + 1)(k + 1) -d + 1
Eq(76)

leads to

(d + 1)(k + 1) ((d + 1)(k + 1) + 2)((d + 1)(k + 1) + 1) a (d+1)(k+1) - (d + 1)k ((d + 1)k + 2)((d + 1)k + 1) a (d+1)k = = -A 1 d + dA 3 -A 3 ((d + 1)k + d + 2) d + 2 A 1 -A 3 (d + 1)k + d + 3 + A 2 d + A 3 ((d + 1)k + 2) d + 2 (d + 1)k + d + 3 - 1 (d + 1)k + d + 2 k-2 l=1 a (d + 1) (k + 1 -l) -d + 1 - 1 (d + 1) (k + 1) -d + 1 k+1 l=2 a (d + 1) l + 1 = -A 1 d + dA 3 -A 3 ((d + 1)k + d + 2) d + 2 A 1 -A 3 (d + 1)k + d + 3 + A 2 d + A 3 ((d + 1)k + 2) d + 2 (d + 1)k + d + 3 - 1 (d + 1)k + d + 2 k-1 l=2 a (d + 1) (l + 1) -d + 1 - 1 (d + 1) (k + 1) -d + 1 k+1 l=2 a (d + 1) l + 1 = B 1 k + d+3 d+1 + B 2 k + d+2 d+1 + B 3 k + 2 d+1 + a (d + 1) 2 2 k + d+3 d+1 - 1 k + d+2 d+1 Ψ k + 2 d + 1 - a (d + 1) 2 1 k + 2 d+1 Ψ k + 2 + 1 d + 1 , ( 77
)
where the Digamma function Ψ(x) as well as the short notations

B 1 = 2A 1 -A 3 d+1 -2a (d+1) 2 Ψ 2 + 2 d+1 , B 2 = -dA 1 +(d-1)A 3 d(d+1) + a (d+1) 2 Ψ 2 + 2 d+1 and B 3 = dA 2 +A 3 d(d+1) + a (d+1) 2 Ψ 2 + 1 d+1 are introduced. Taking U k = (d + 1)k ((d + 1)k + 2)((d + 1)k + 1) a (d+1)k and G(k + 1) = B 1 k + d+3 d+1 + B 2 k + d+2 d+1 + B 3 k + 2 d+1 + a (d + 1) 2 2 k + d+3 d+1 - 1 k + d+2 d+1 Ψ k + 2 d + 1 - a (d + 1) 2 1 k + 2 d+1 Ψ k + 2 + 1 d + 1 . ( 78
)
then (77) can be written in compact form as

U k+1 -U k = G(k + 1).
The later recurrence is easily solved to give

U k = U 3 + k j=4 G(j).
By using the formula Ψ(j + 1) = Ψ(j) + 1/j and the relations [13, Theorems 3.1 and 3.2]

k l=0 Ψ(l + α) l + β + k l=0 Ψ(l + β + 1) l + α = Ψ(k + α + 1)Ψ(k + β + 1) -Ψ(α)Ψ(β), ( 79
) k j=0 Ψ(j + β) j + β = 1 2 Ψ ′ (k + β + 1) -Ψ ′ (β) + Ψ(k + β + 1) 2 -Ψ(β) 2 , ( 80
)
we obtain

U k = (d + 1)k ((d + 1)k + 2)((d + 1)k + 1) a (d+1)k = B 1 Ψ k + d + 3 d + 1 + B 2 Ψ k + d + 2 d + 1 + B 3 Ψ k + 2 d + 1 + 2a (d + 1) ((d + 1)k + 2) + a (d + 1) 2 Ψ ′ k + d + 3 d + 1 + Ψ k + d + 3 d + 1 2 - a (d + 1) 2 Ψ k + 2 d + 1 Ψ k + d + 2 d + 1 + δ 2 , ( 81
)
where

δ 2 = a (d + 1) 2 Ψ 2 d + 1 Ψ d + 2 d + 1 -B 1 Ψ d + 3 d + 1 -B 2 Ψ d + 2 d + 1 -B 3 Ψ 2 d + 1 - a d + 1 - a (d + 1) 2 Ψ ′ d + 3 d + 1 - a (d + 1) 2 Ψ d + 3 d + 1 2 -G(1) -G(2) -G(3) + U 3 .
From (81) we deduce the asymptotic behaviour of a (d+1)k as k → ∞:

a (d+1)k = δ 1 k + 3 d + 1 + 2 (d + 1) 2 k + a (d + 2) (d + 1) 2 + 1 2 a (d + 4) (d + 1) 3 k + 1 6 ad (d + 2) (d + 1) 4 k 2 - 1 4
ad (d + 2) (d + 1) 5 k 3 + • • • ln(k) + δ 2 (d + 1)k + δ 3 + δ 4 k + δ 5 k 2 + δ 6 k 3 + • • • (82)
where coefficients δ i are defined by (higher terms are omitted)

δ 1 = (d + 1)(B 1 + B 2 + B 3 ), δ 3 = d 2 + d + 4 d + 1 B 1 + d 2 + 3 d + 1 B 2 - d -3 d + 1 B 3 + 3 δ 2 + 3 d + 1 a, δ 4 = 5 d 2 + 4 d + 47 6 (d + 1) 2 B 1 + 8 d 2 + d + 41 6 (d + 1) 2 B 2 - d 2 + 8 d -41 6 (d + 1) 2 B 3 + 2 d + 1 δ 2 + 9 d + 13 2 (d + 1) 3 a, δ 5 = d 2 + 2 d + 9 6 (d + 1) 3 B 1 - d 3 + 2 d 2 -9 6 (d + 1) 3 B 2 + d 2 + 2 d + 9 6 (d + 1) 3 B 3 + d 3 + 5 d 2 -2 d + 6 12 (d + 1) 4 a, δ 6 = d 2 + 2 d -19 60 (d + 1) 2 B 1 + d 3 + 18 d 2 + 13 d -19 60 (d + 1) 3 B 2 + d 2 + 2 d -19 60 (d + 1) 2 B 3 - 7 d 3 + 20 d 2 -11 d -8 24 (d + 1) 5 a, . . .
At this level we should remark that lim k→∞ a 2k = ∞ for all δ i , i = 1, 2, 3, ..., since a = 0. Recall that

c d n = T 1 d+1 (n -d + 1) an a n-d -(n -d) ,
for n ≥ d, then the equation (74) can be written as

a (d+1)k+d b + T 1 d + 1 (d + 1)k + d + 2 a (d+1)k = φ(k), (83)
where

φ (k) = T 1 d + 1 ((d + 1)k)((d + 1)k + d + 2) (d + 1)k + 1 -bA 2 + bA 3 (d + 1)k + d + k-1 l=2 ab (d + 1)l + 1 ,
and the equation ( 75) can be written

a (d+1)(k+1) b + T 1 d + 1 (d + 1)(k + 1) + 2 a (d+1)k+1 = φ(k), (84)
where

φ (k) = T 1 d + 1 ((d + 1)(k + 1) + 2)((d + 1)k + 1) (d + 1)k + 2 + bA 1 + bA 3 (d + 1)(k + 1) + k l=3 ab (d + 1)l -d + 1
.

From ( 83) and (84) we have

1 a (d+1)(k+1)+d = 1 φ (k + 1) b + T 1 d + 1 (d + 1)(k + 1) + d + 2 a (d+1)(k+1) (85) and 1 a
(d+1)k+1 = d + 1 T 1 1 (d + 1)(k + 1) + 2 φ(k) a (d+1)(k+1) -b , (86)
which give an explicit formula for a (d+1)(k+1)+d and a (d+1)k+1 .

• If we suppose lim k→∞ a (d+1)k (d+1)k = ∞, then from (85) and ( 86) we deduce on one side lim k→∞

(d + 1)(k + 2) a (d+1)(k+1)+d = lim k→∞ (d + 1)(k + 2) φ (k + 1) b + T 1 d + 1 (d + 1)(k + 1) + d + 2 a (d+1)(k+1) = (d + 1)b T 1 (87)
and

lim k→∞ (d + 1)(k + 1) + 2 a (d+1)k+1 = lim k→∞ d + 1 T 1 φ(k) a (d+1)(k+1) -b = - (d + 1)b T 1 . (88)
On the other side, for n = (d + 1)k + 2d + 1, (41) reads

(d + 1)T 2 T 2 1 1- (d + 1)k (d + 1)k + 1 a (d+1)k a (d+1)(k+1)+d = (d + 1)(k + 2) a (d+1)(k+1)+d - 2((d + 1)(k + 1) + 1) a (d+1)(k+1) + (d + 1)k + 2 a (d+1)k+1 . (89)
Under the assumption T 2 = 0, (89) admits the limit ∞ = 0, as k → ∞, which exhibit a contradiction.

• Now if lim k→∞ a (d+1)k (d+1)k = η 1 = 0, then from (85) and (86) we have lim k→∞

(d + 1)(k + 2) a (d+1)(k+1)+d = lim k→∞ (d + 1)(k + 2) φ (k + 1) b+ T 1 d + 1 (d + 1)(k + 1) + d + 2 a (d+1)(k+1) = (d + 1)b T 1 + 1 η 1 := η 2 , (90) and lim k→∞
(d + 1)(k + 1) + 2 a (d+1)k+1 = lim k→∞ d + 1 T 1 φ(k) a (d+1)(k+1) -b = 1 η 1 - (d + 1)b T 1 = 2 η 1 -η 2 . (91)
By taking the limit in (89) we obtain,

(d + 1)T 2 T 2 1 (1 -η 1 η 2 ) = η 2 - 2 η 1 + 2 η 1 -η 2 = 0.
If η 2 = 0, we have the contradiction T 2 = 0. But if η 2 = 0, then η 2 = 1/η 1 . From (90) we get (d + 1)b/T 1 + 1/η 1 = η 2 , which gives the contradiction b = 0.

• Finally if lim k→∞ a (d+1)k (d+1)k = 0, then according to (82), we have δ 1 = δ 2 = 0 and

a (d+1)k = ((d + 1)k + 2)((d + 1)k + 1) (d + 1)k B 1 Ψ k + d + 3 d + 1 + B 2 Ψ k + d + 2 d + 1 -(B 1 + B 2 )Ψ k + 2 d + 1 + 2a (d + 1) ((d + 1)k + 2) + a (d + 1) 2 Ψ ′ k + d + 3 d + 1 + Ψ k + d + 3 d + 1 2 - a (d + 1) 2 Ψ k + 2 d + 1 Ψ k + d + 2 d + 1 , = a (d + 2) (d + 1) 2 + 1 2 a (d + 4) (d + 1) 3 k + 1 6 ad (d + 2)
(d + 1) 4 k 2 - 1 4
ad (d + 2) (d + 1) 5 k 3 + • • • ln(k)+δ 3 + δ 4 k + δ 5 k 2 + δ 6 k 3 + • • • .( 92
)
Let write the equation (89) as

(d + 1)T 2 T 2 1 1- (d + 1)k (d + 1)k + 1 a (d+1)k a (d+1)(k+1)+d - (d + 1)(k + 2) a (d+1)(k+1)+d + 2((d + 1)(k + 1) + 1) a (d+1)(k+1) - (d + 1)k + 2 a (d+1)k+1 = 0. ( 93
)
After multiplying the both sides of the equation (93) by φ(k + 1)a (d+1)(k+1) and using (92), ( 85) and (86), we get, as k → ∞,

λ 1 ln(k) 2 + λ 2 ln(k) + λ 3 + • • • = 0 (94)
where

λ 1 = - (d + 2) b (d + 1) 3 T 2 1 T 2 + b 2 (d + 1) 2 T 1 a 2
and

λ 2 = 2b(d + 2)a 2 (d + 1) 3 T 2 1 Ψ 2d + 3 d + 1 -3 (d + 3)ba 2 (d + 1) 2 T 2 1 + 2 bA 2 (d + 2) + T 1 (d + 3) (d + 1) 2 T 2 1 a T 2 - 2b 2 a 2 (d + 1) 2 T 1 Ψ 2d + 3 d + 1 + 3 b 2 a 2 (d + 1)T 1 -2 (bA 2 + T 1 )ba (d + 1)T 1 .
So, we must have λ i = 0, i = 1, 2, 3, .... As a = 0 and b = 0, from λ 1 = 0 we get Case 3: For every k 0 ≥ 3, there exists infinitely many k, κ ≥ k 0 such that: D k = 0 and D κ = 0. We take k 1 and k 2 ,

T 2 = d + 1 d + 2 bT 1 , (95)
k 1 = k 2 , with D k 1 = 0, D k 1 +1 = 0, D k 2 = 0 and D k 2 +1 = 0 to get from (45) that a n - n -k 1 (d + 1) -2d -1 n -k 1 (d + 1) -2d a n-k 1 (d+1)-2d-1 and a n - n -k 2 (d + 1) -2d -1 n -k 2 (d + 1) -2d a n-k 2 (d+1)-2d-1 (97)
are two rational functions of n. Consequently, an analogous reasoning to that of Lemma 4 completes the proof.

In the next subsection we give some expressions concerning the sequence {γ d n } n≥d and the power series F (t). The later is expressed by hypergeometric series (25).

3.1

Expressions for γ with

γ d n = T 1 d + 1 (n -d + 1) α n-d+1 α n+1 -(n -d) α n-d α n , for n ≥ d, (99)
and for n = dm + r, m ≥ 1, 1 ≤ r ≤ d we have

γ d dm+r = T 1 d + 1 (dm + r)!(β r (m -r -1) + (d + 1)b r ) (d(m -1) + r)! r l=1 (β l m + b l ) d l=r (β l (m -1) + b l ) (100) with γ d d = T 1 d!/ d l=1 b l , b l = (l + 1)α l+1 /α l and β l = b d+l -b l , 1 ≤ l ≤ d.
Proof: The equation ( 39) is

c d n = α n α n-d γ d n = T 1 d + 1 (n -d + 1) a n a n-d -(n -d) , for n ≥ d. ( 101
)
As a n = α n /α n+1 we get (99).

According to Theorem 5 we have R(t) = T 1 t d+1 /(d + 1). So, by Corollary 5 we obtain

c d n = α n α n-d γ d n = T 1 d + 1 (n + 1) b n-d b n -(n -d) , for n ≥ d + 1, (102)
γ d n = T 1 d + 1 α n-d ((n + 1)b n-d -(n -d)b n ) α n b n , for n ≥ d + 1. ( 104
)
We calculate α n b n /α n-d by using the relation

α n = b n-1 α n-1 /n = n-1 l=0 b l /n! to find α n b n α n-d = (n -d)! n! n l=0 b l n-d-1 l=0 b l = (n -d)! n! n l=n-d b l . ( 105
)
Now for n = md + r, we can write

Proposition 4 If d l=1 β l = 0 then F (t) = 1 + F 1 (t) + F 2 (t),
where

F 1 (t) = α d t d d+1 F d   1, b d β d , b l+d β l d-1 l=1 l+d d d l=1 ; d l=1 β l t d d   (108)
and

F 2 (t) = d-1 r=1 α r t r d F d-1   b l+d β l r-
F 1 (t) = α 1 βd   d F d-1   βd β d , b l β l d-1 l=1 l d d-1 l=1 ; d l=1 β l t d d   -1   . (110)
Proof:

Recall that α n = n-1 l=0 b l /n!. If d l=1 β l = 0, then for n = md + r and using the expression of b md+r , we obtain

md+r-1 l=0 b l = b 0 b 1 b 2 • • • b r-1 b r b r+1 • • • b d+r-1 • • • b (m-1)d+r-1 b (m-1)d+r • • • b md b md+1 • • • b md+r-1 = (b 0 b 1 b 2 • • • b r-1 )(b r b r+d • • • b r+(m-1)d )(b r+1 • • • b r+1+(m-1)d ) • • • (b d+r-1 b 2d+r-1 • • • b md+r-1 ) = r-1 l=0 b l d+r-1 l=r b l b l+d • • • b l+(m-1)d = r!α r d l=r b l b l+d • • • b l+(m-1)d d+r-1 l=d+1 b l b l+d • • • b l+(m-1)d = r!α r d l=r β l m d l=r b l β l m r-1 l=1 β l m r-1 l=1 1 + b l β l m = r!α r d l=1 β l m d l=r b l β l m r-1 l=1 b l+d β l m . (111)
We have also, for m ≥ 0 and 0 ≤ r ≤ d -1, the expressions [6, Lemma So,

α md+r = α r r-1 l=1 b l+d β l m d l=r b l β l m m! d-1 l=r+1 l d m d+r l=d+1 l d m d l=1 β l 1 d d m , for 1 ≤ r ≤ d -1, (112)
α md+d = α d d-1 l=1 b l+d β l m b d β d m d l=1 1 + l d m d l=1 β l 1 d d m (113)
and, if βd = 0,

α md+d = α 1 βd d-1 l=1 b l β l m+1 βd β d m+1 (m + 1)! d-1 l=1 l d m+1 d l=1 β l 1 d d m+1 . (114)
Now, expanding F (t) as

F (t) = n≥0 α n t n = 1 + m≥0 α md+d t md+d + d-1 r=1 m≥0
α md+r t md+r , the expressions (108), ( 109) and (110) follow from (112), ( 113) and (114), respectively.

Example 1 If d = 1 then R(t) = T 1 t 2 /2, b n = (b 2 -b 1 )n + 2b 1 -b 2 = β 1 n + β1 , for n ≥ 1, and F (t), γ d n have the expressions: 1) If β 1 = 0 we have F (t) ≡ F β 1 (t) = 1 + α 1 t 2 F 1 1, b 1 β 1 2 ; β 1 t (115) with γ 1 1 = T 1 /b 1 and for n ≥ 2, γ 1 n = T 1 2 n(β 1 (n -1) + 2 β1 ) (β 1 n + β1 )(β 1 (n -1) + β1 )
.

The limiting case is lim

β 1 →0 F β 1 (t) = 1 + α 1 t 1 F 1 1 2 ; b 1 t = 1 + α 1 b 1 e b 1 t -1 (116) with γ 1 n = T 1 n/b 1 for n ≥ 1. 2) If β1 β 1 = 0, [2, 14], F (t) ≡ F β 1 , β1 (t) = 1 + α 1 β1 (1 -β 1 t) -β1 β 1 -1 (Ultraspherical polynomials) (117)
with

γ 1 n = T 1 2 n(β 1 (n -1) + 2 β1 ) (β 1 n + β1 )(β 1 (n -1) + β1 ) , for n ≥ 1.
The limiting cases are lim

β 1 →0 F β 1 , β1 (t) = 1 + α 1 b 1 e b 1 t -1 (Hermite polynomials) (118)
and lim

β1 →0 F β 1 , β1 (t) = 1 - α 1 β 1 ln(1 -β 1 t) (Chebyshev polynomials of the first kind). (119)
Remark that (117) and (119) are special cases of (115) for β1 β 1 = 0 and β1 = 0,i.e. β 1 = b 1 , respectively. Also, (116) is exactly (118), since in this case β 1 → b 1 .

Example 2 For d ≥ 1 we take b n = αn + β for all n ≥ 1. So, from (103) we have β r = dα and, of course, b r = αr + β, for 1 ≤ r ≤ d. Thus, for αβ = 0, γ d n becomes

γ d n = T 1 α -d-1 d + 1 n!(α(n -d) + (d + 1)β) (n -d)!(n + β α -d) d+1 , n ≥ d, (120)
and for F (t), since α n = n-1 l=0 b l /n!, we obtain (see [START_REF] Anshelevich | A characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind[END_REF] for calculations) A. The first representation by (108) and (109).

F (t) ≡ F α,β (t) = 1 + α 1 β (1 -αt) -β/α -1 . (121
If β 1 β 2 = 0, then

F (t) = 1 + α 2 t 2 3 F 2 1, b 2 β 2 , b 3 β 1 3 2 , 2 ; β 1 β 2 t 2 2 + α 1 t 2 F 1 b 1 β 1 , b 2 β 2 3 2 ; β 1 β 2 t 2 2 . ( 130
)
The limiting cases are obtained when: β 1 → 0, β 2 → 0 or (β 1 , β 2 ) → (0, 0). B. The second representation, by (109) and (110), with its limiting cases.

1. If β2 β 1 β 2 = 0 we have

F (t) = 1 + α 1 β2 2 F 1 β2 β 2 , b 1 β 1 1 2 ; β 1 β 2 t 2 2 -1 + α 1 t 2 F 1 b 1 β 1 , b 2 β 2 3 2 ; β 1 β 2 t 2 2 . ( 131
)
2. If β 1 → 0:

F (t) = 1 + α 1 β2 1 F 1 β2 β 2 1 2 ; b 1 β 2 t 2 2 -1 + α 1 t 1 F 1 b 2 β 2 3 2 ; b 1 β 2 t 2 2 . ( 132
)
with

γ 2 2m = T 1 3b 1 2m(2 m -1)(β 2 (m -1) + 3 β2 ) (β 2 m + β2 )(β 2 (m -1) + β2 ) , m ≥ 1 ( 133
)
γ 2 2m+1 = T 1 b 1 (2m)(2m + 1) β 2 m + β2 , m ≥ 1, (134)
3. If β 2 → 0:

F (t) = 1 + α 1 b 2 1 F 1 b 1 β 1 1 2 ; b 2 β 1 t 2 2 -1 + α 1 t 1 F 1 b 1 β 1 3
F (t) = 1 + α 2 t 2 3 F 2 1 1 1 + b 1 β 1 2 3 2 ; β 1 b 2 t 2 2 + α 1 t 2 F 1 1 b 1 β 1 3 2 ; β 1 b 2 t 2 2 . ( 138
)
with γ 2 2 = 2T 1 /(b 1 b 2 ),

γ 2 2m = T 1 3b 2 2(2m -1) (β 1 (m -1) + b 1 )
, m ≥ 2 (139) 5. If β 1 → 0 and β 2 → 0:

γ 2 2m+1 = T 1 3b 2
F (t) = 1 + α 1 b 2 0 F 1 - 1 2 ; b 1 b 2 t 2 2 -1 + α 1 t 0 F 1 - 3
F (t) = 1 + α 2 t 2 2 F 2 1 1 2 3 2 ; b 1 b 2 t 2 2 + α 1 t 1 F 1 1 3 2 ; b 1 b 2 t 2 2 . ( 143
)
with γ 2 2 = 2T 1 /(b 1 b 2 ),

Lemma 4

 4 d+1)m-d = 0 which represents the condition of corollary 3 with κ = (d + 1)m ≥ 3(d + 1) and therefore gives R 2d+2 = T 2 = 0. If T κ = T m = 0 for some κ = m ≥ 3, then T 2 = 0.

  and by replacing a = T 2 /b 2 and (95) in the equation λ 2 = 0 we obtaind -1 (d + 2) 3 T 1 2 = 0. (96) If d = 1, (96) gives T 1 = 0 which is a contradiction. The case d = 1 is already treated, [14, Theorem 1].

d n and F (t) Proposition 3

 3 The d-symmetric d-PS, {P n }, generated by (1) satisfies xP n (x) = P n+1 (x) + γ d n P n-d (x), n ≥ 0, P -n (x) = 0, 1 ≤ n ≤ d, and P 0 (x) = 1 (98)

  with b n = (n + 1)/a n = (n + 1)α n+1 /α n andb md+r = β r m + b r , for m ≥ 0, 1 ≤ r ≤ d,(103)whereβ r = b d+r -b r , for 1 ≤ r ≤ d.The equation (102) gives

  l (m -1) + b l ) r l=1 (β l m + b l ) (106) and (n + 1)b n-d -(n -d)b n = (md + r + 1)b (m-1)d+r -(md + r -d)b md+r = (md + r + 1)(β r (m -1) + b r ) -(md + r -d)(β r m + b r ) = β r (m -r -1) + (d 1)b r . and (100) follows by combining (105), (106) and (107).

  βd := b d -β d = 2b d -b 2d = 0, then F 1 (t) in (108) can be written as

(

  md + r)! = r!m!d dm d

Remark 4

 4 In proposition 4 two expressions of F (t) are given. The first, when d l=1 β l = 0, Equations (108) and (109), from which we can deduce the other limiting cases by tending to zero at least a constant β r , 1 ≤ r ≤ d. So, we can enumerate 2 d expressions of F (t) similar to that given in [6, Theorem 3.1]. The second, when d l=1 β l = 0 and βd = b d -β d = 0, Equations (110) and (109), seems to be a new representation of F (t). Similarly, the other limiting cases can be obtained by tending to zero at least a constant β r , 1 ≤ r ≤ d, and βd . So, in this second representation, we can enumerate 3 • 2 d-1 (= 2 d + 2 d-1 ) expressions of F (t). We note that, the resulting 2 d-1 expressions when βd → 0, i.e. β d → b d , are special cases of the first representation when β d = b d . See the illustrative examples given below.

)Fdbt d+1 1 -Example 3 3 2m( 2 m

 1332 Let λ = β/α. Then for T 1 = α d+1 (d + 1) -d and with the change of variable t → (d + 1)t/α in the generating function 1 -α(xt -T 1 t d+1 /(d + 1))-λ , we meet the Humbert polynomials[START_REF] Humbert | Some extensions of Pincherele's polynomials[END_REF] generated by1 -(d + 1)xt + t d+1 -λ . For d = 1we have the ultraspherical polynomials.The limiting cases are 1. α → 0 and β = 0:lim α→0 n = T 1 β -d n!/(n-d)!, for n ≥ d. In the generating function exp β xt -T 1 t d+1 /(d + 1) , with T 1 = β d ((d + 1)d!) -1 and the change of variable t → t/β, we find the generating function exp xt -(d + 1) -2 t d+1 /d! of the Gould-Hopper polynomials [10] . 2. β → 0 and α = 0 lim β→0 F α,β (t) = 1 -α 1 α ln(1 -αt) (123) with γ d d = T 1 α -d and γ d n = T 1 α -d /(d + 1) for n ≥ d + 1. Let b = T 1 α -d /(d + 1). Then by the shift n → n + d in (98), these polynomials satisfyP n+d+1 (x) = xP n+d (x) -bP n (x), n ≥ 1, P n (x) = x n , 0 ≤ n ≤ d, and P d+1 (x) = xP d (x) -(d + 1)bP 0 (x), (124)where we recognise the monic Chebyshev d-OPS of the first kind generated by (see[START_REF] Cheikh | d-orthogonal polynomial sets of Chebyshev type[END_REF] Theorem 5.1])1 -dbt d+1 1 -xt + bt d+1 = n≥0 P n (x)t n . xt + bt d+1 -1 dt = -ln 1 -xt + bt d+1 . (126)Then, by changing the variable t → αt, multiplying by α 1 /α and adding 1 in (126), we get the generating function (with F (t) as in (123)),1 -α 1 α ln 1 -α xt -T 1 d + 1 t d+1 = 1 + n≥1 α 1 αn P n (x)t n .(127) For d = 2 we have R(t) = T 1 t 3 /3 and from (100) we get, for m ≥ 1, the two expressions+ 1)(2 m + 1)(β 2 m + 3 β2 ) (β 1 m + b 1 )(β 2 m + b 2 )(β 2 m + β2 ) + 1)(β 1 (m -2) + 3 b 1 ) (β 1 m + b 1 )(β 1 (m -1) + b 1 )(β 2 m + β2 ) ,(129) with γ 2 2 = 2T 1 /(b 1 b 2 ), β 1 = b 3 -b 1 , β 2 = b 4 -b 2 and β2 = 2b 2 -b 4 . We enumerate the following forms of F (t):

2 ; b 2 β 1 t 2 2 . 2 2m( 2 m) 4 .

 22224 + 1)(β 1 (m -2) + 3 b 1 ) (β 1 m + b 1 )(β 1 (m -1) + b 1 ), m ≥ 1, (137If β2 → 0:

2( 2 m

 2 + 1)(β 1 (m -2) + 3 b 1 ) (β 1 m + b 1 )(β 1 (m -1) + b 1 ), m ≥ 1. (140)Clearly (138) is (130) with β 2 = b 2 .

2 ; b 1 b 2 t 2 2 6 .

 226 If β 1 → 0 and β2 → 0:

Email: m mesk@yahoo.fr

Email: m b zahaf@yahoo.fr

Acknowledgements: We would like to thank Dr. Yanallah Abdelkader for precious help and useful discussions.

Considering Lemma 4, we can choose k ≥ k 0 such that T k = 0 for k ≥ k -1. Let define, for k ≥ k, Dk = D k T k-1 T k and Ēk (n) be the equation ( 45) divided by T k-1 T k T k+1 . By making the operations

we get, for k ≥ k + 1, the equation

where W k,l is independent of n and

Similarly, by the operations

and the shift n

where W k,l is independent of n. Now, for k = κ ≥ k + 1, the equations ( 46) and (48) give, respectively,

and

If Dk = Dκ for some k = κ ≥ k + 1, then by ( 49) and ( 50) we can eliminate a n-d-1 to get that a n is a rational function of n. So, by Corollary 4, we have T 2 = 0.

If Dk = D for k ≥ k + 1, then (46) and (48) become, respectively,

and

The combinations ((n + d + 2)Eq(52) -(n + d + 1)Eq(51)) /(d + 1) and ((n -2d -1)Eq(52) -(n -2d -2)Eq(51))/(d + 1) give, respectively,


Jamal Bouoiyour
email: jamal.bouoiyour@univ-pau.fr

Refk Selmi

Are Trump and Bitcoin Good Partners?

Keywords: 2016 U.S. presidential elections, Trump's win, U.S. stock market, Bitcoin price, safe haven

come

Introduction

Bitcoin was created in 2009 by an anonymous programmer under the pseudonym Satoshi Nakamoto and has since achieved a widest level of international recognition. Given the growing attention given to Bitcoin, researchers have revolved around its properties by asking several questions. One can cite: Is Bitcoin an alternative currency? Is it a speculative bubble? Is it a safe haven? Is it a hedge? Is it a portfolio diversifier? etc…

The majority of studies supported that Bitcoin is likely to be a speculative bubble rather than a safe haven or a hedge (Bucholz et al. 2012;[START_REF] Kristoufek | Bitcoin meets Google Trends and Wikipedia: Quantifying the relationship between phenomena of the Internet era[END_REF]Ciaian et al. 2014;Yermack 2014;Bouoiyour et al. 2015;Bouoiyour and Selmi 2015). Some researchers attributed this great speculation to the fact that there is no guarantee of repayment at any time [START_REF] Kristoufek | Bitcoin meets Google Trends and Wikipedia: Quantifying the relationship between phenomena of the Internet era[END_REF]Yermack 2014). Bitcoin is also not yet accepted as a payment system across large markets and does not have an underlying value derived neither from consumption nor production process such as the precious metals including gold (Glaster et al. 2014). In addition, being a virtual currency, Bitcoin is highly sensitive to cyber-attacks, which can easily destabilise the Bitcoin system and then generate excessive price volatility [START_REF] Ciaian | The economics of BitCoin price formation[END_REF]. Another potential element that may explain the speculative attitude of Bitcoin is its strong dependence to media coverage. Accordingly, Lee (2014) showed that the alteration of positive and negative news contributed to high Bitcoin price cycles. Moreover, exponents of Bitcoin's virtues indicated that as it is an alternative currency and transaction tool that does not depend on a network of financial institutions, users are insulated against the untoward risks. This seems relevant when we consider that Bitcoin was created with the onset of the global financial collapse when governments and financial institutions lose public trust. This has give weight to compare Bitcoin to gold, as gold is perceived, in theory, as a hedge and safe haven to protect against similar risks. Baur and Lucey (2010) and [START_REF] Baur | Is gold a safe haven? International evidence[END_REF] claimed that there exists a significant linkage among gold price and financial assets. They defined a safe haven and a hedge depending on the sign and the significance of the coefficients associated to the gold price. The property of a safe-haven asset is to deliver or retain its value during times of crisis. So, the primary characteristic of a safe haven is the specific period in which the return is non-negative, whereas hedge has to be negatively correlated (or uncorrelated). In this context, Dyhberg (2015) tried to test the hedging capabilities of Bitcoin by addressing whether Bitcoin is a virtual gold. The study suggests that Bitcoin can serve as a hedge against stocks over shortest horizons. It concludes that Bitcoin possess hedging characteristics as gold and can be incorporated in a portfolio to mitigate the harmful effects of sudden shocks.

Even though the literature has largely documented how precious metals (with large extent gold) can act as a safe haven during financial turmoil [START_REF] Baur | Is gold a Hedge or Safe Haven? An Analysis of Stocks Bonds and Gold[END_REF]Bouoiyour and Selmi 2016), it ignored the role that may be played by Bitcoin as a hedging investment and a portfolio diversifier. Bitcoin is an alternative to mainstream currencies and the economy [START_REF] Bouri | On the hedge and safe haven properties of Bitcoin: Is it really more than a diversifier[END_REF]. If traders and investors lose trust to currencies or to the whole economy, they may resort to Bitcoin. This is one of the potential causes behind the consideration of Bitcoin as a -digital gold‖ [START_REF] Popper | Digital gold: The untold story of Bitcoin[END_REF]. Despite its relevance, research examinaning Bitcoin as a diversifier, hedge, or safe heaven is extremely limited. To our best knowledge, there are only two studies that have attempted to answer this question [START_REF] Bouri | On the hedge and safe haven properties of Bitcoin: Is it really more than a diversifier[END_REF]Dyhberg 2015). This study tries to fill this gap by addressing whether Bitcoin join gold in hedging and safe haven status during the political uncertainty surrounding the 2016 U.S. presidential elections.

Financial markets had widely priced in a win for Clinton, who they viewed as a better short-run outcome because she represented few unknowns and thus less uncertainty. In response to the Trump's stunning triumph, the asset markets around the world plunged markedly as investors were concerned about evolving volatility. This has led to a trend towards questioning the effectiveness of standard economic and financial structures which govern the conventional monetary and financial system. Here, the digital currency (in particular, Bitcoin) is leading the charge by providing a completely decentralized secure alternative to fiat currencies during times of economic and geopolitical unrest. The price of Bitcoin starts 2017 as the top currency; it exceeds $ 1000 in First Day of 2017 trading for the first time on Coindesk Bitcoin Price Index since early January 2014.

Because the interaction of complex phenomena like Bitcoin and asset markets may mask the regularities we would like to identify, we apply a new data analysis tool, namely Ensemble Empirical Mode Decomposition (EEMD), which decomposes the Bitcoin price and the U.S. stock price index (in particular, S&P 500) into a scale-on-scale basis and at each scale it is estimated the correlation. The motivations behind the use of this technique arise in the desire to extract intrinsic characteristics inherent to the time series. In doing so, we show that hedge and safe-haven properties of Bitcoin and precious metals for U.S. stock market is timevarying. While Bitcoin serves as a weak safe haven (negative correlation) in shorter time scales, and as a hedge in the medium-and the long-run (insignificant correlation). We show also that gold and silver lost their hedge and safe haven properties as the correlation between precious metals and U.S. stock price is declining from one scale to another.

The outline of the paper is as follows: Section 2 describes our methodology. Section 3 reports the results. Section 4 discusses the findings and concludes.

Methodology and data

The traditional time series analysis tools usually rely on Fourier transforms in one way or another. Nevertheless, according to [START_REF] Huang | The empirical mode decomposition method and the Hilbert spectrum for non-stationary time series analysis[END_REF], the Fourier transform might prompt inaccurate information owing essentially to the nature (in the time domain) of the transform. Wavelets have a problem of shift variance. Precisely, if the start point varies, by for example dropping the initial point, the wavelet transform may reveal distinct outcomes. However, the EMD method makes no assumption about linearity or stationarity and the intrinsic mode functions (IMFs) are often easily depicted3 . A signal can be adaptively disentangled into a sum of finite number of zero mean oscillating components having symmetric envelopes defined by the local maxima and minima. The EMD is based on the sequential extraction of energy associated with distinct frequencies ranging from high fluctuating components (short-run) to low fluctuating modes (long-run). With the Hilbert transform, the IMF prompts instantaneous frequencies as functions of time that help to properly identify imbedded structures. The EMD aims at transforming the time series to hierarchical structure by means of the scaling transformations. In brief, it quantifies the changeability captured via the oscillation at different scales and locations.

In practice, the IMFs are decomposed by determining the maxima and minima of series ) (t x , generating then its upper and lower envelopes ) ( ( min t e and ) ( m ax t e

) with cubic spline interpolation. We first measure the mean ) (t m for different points from upper and lower envelopes:

2 / )) ( ) ( ( ) ( m ax m in t e t e t m   (1)
We decompose the mean of the time series under study to identify the difference

) (t d between ) (t x and ) (t m : ) ( ) ( ) ( t x t m t d  
(2)

We present ) (t d as the i th IMF and we replace

) (t x with the residual ) ( ) ( ) ( t d t x t r  
.

If not, we replace

) (t x with ) (t d .
Then, we connect the local maxima with the upper envelope and the minima with the lower one. When residue meets the conditions that the number of zerocrossings and extrema do not differ by more than one and when the residue (r) becomes a monotonic function and data cannot be extracted into further intrinsic mode functions [START_REF] Wu | A study of the characteristics of white noise using the empirical mode decomposition method[END_REF], the original time series can be denoted as the sum of some IMFs and a residue:

) ( ) ( ) ( 1 t r t c t X N j j     (3)
The EMD technique [START_REF] Huang | The empirical mode decomposition method and the Hilbert spectrum for non-stationary time series analysis[END_REF] is illustrated more simply in Figure 1. The decomposition of the signal into IMFs is carried out as follows: After determining the positive peaks (maxima) and negative peaks (minima) of the original signal, we construct the lower and the upper envelopes of the signal by the cubic spline method (red). In addition, we measure the mean values (blue) by averaging the upper envelope and the lower envelope. Besides, we subtract the mean from the original signal to find the first intrinsic mode function (IMF1). Then, we calculate the first residual component by subtracting IMF1 component from the original signal. Finally, we repeat the steps above until the final residual component becomes a monotonic function and no more IMFs can be extracted. As effective as EMD proved to be, this new technique leaves some difficulties unresolved. One of the major shortcomings of the EMD is the mode mixing which is the result of signal intermittency. This may generate serious aliasing in the time-frequency distribution, and then the intrinsic modes might devoid of physical meaning. The Ensemble Empirical Mode Decomposition (EEMD), proposed by [START_REF] Wu | Ensemble empirical mode decomposition: A noise-assisted data analysis method[END_REF] as a truly noise-assisted data analysis, permits to avoid this problem. It defines the extracted IMFs as the means of an ensemble of trials. Each trial disentangles the signal and adding a finite amplitude white noise. The addition of noise could help data analysis in the EMD method and thus could minimize or overcome the mode-mixing problem. Figure 2 illustrates the transition from EMD to EEMD consisting of adding a White Gaussian Noise (AWGN), where the std is the noise amplitude in terms of standard deviations, and Ne is the ensemble number. To control for mode-mixing, the EEMD treats the EMD as basic function and repeats Ne times with distinct AWGNs.

Figure 2. EMD versus EEMD

After the partition of the signal into a set of different frequency components by means of EEMD, a correlation analysis between Bitcoin price and U.S. stock price index and potential control variables is conducted. The model to be estimated is given by:

t t t t t t t WTI silver gold BP SPI SPI               1 (4)
Where SPI is the S&P 500 Stock Price Index; The S&P 500 Stock Price Index (SPI) covers the performance of 500 largest capitalization stocks, and is sourced from DataStream of Thomson Reuters; SPI t-1 : the lagged S&P 500 stock price that may reflect the influence of some potential explanatory variables not included here due to the unavailability of daily frequency data; BP is the Bitcoin price in US dollars collected from CoinDesk at www.coindesk.com/price.The CoinDeskBitcoin Price Index represents an average of Bitcoin prices across leading bitcoin exchanges; gold is the world gold price; silver is the world silver prices. Gold and silver are precious metals that have been largely viewed as safe haven over extreme stock market fluctuations [START_REF] Baur | Is gold a Hedge or Safe Haven? An Analysis of Stocks Bonds and Gold[END_REF]. WTI is the West Texas Intermediate oil price. The WTI has been largely employed in the literature as the benchmark price for global oil markets. The WTI is among the most traded oil on the world markets, and therefore is significantly affected by macro-financial variables. WTI, gold and silver data come from quandl website; t  is the error term.

We focus our analysis on the 2016 US presidential election outcome and address whether Bitcoin acts as a safe haven for U.S. stock price index in the aftermath of Trump's win. The final result of the election was disclosed on Tuesday 08 Nov 2016, which we subsequently view as the announcement day. So, this study uses daily data for the period that spans from 08 Nov 2016 to 15 Feb 2017 (99 observations).

Results

A multi-scale correlation analysis is conducted to assess whether Bitcoin serves a hedge or safe haven for U.S. stock index over the uncertainty surrounding the Trump's victory in U.S. presidential elections. The procedure consists of (1) decomposing the original time series into various intrinsic mode functions, and (2) examining the correlation between Bitcoin price and S&P500 index across the extracted components, and (3) addressing whether Bitcoin joins precious metals in safe-haven status, while controlling for gold and silver prices.

Based on EEMD, the Bitcoin price and S&P500 price index and the explanatory variables (gold and silver prices) are decomposed into five IMFs (Appendix A). The EEMD technique generates itself the modes depending to the data. All the derived IMFs are listed from high to low frequency components. We note that the frequencies and amplitudes of all the IMFs evolve over time. We discuss three scaling components: short-run (IMFs1-2: within one to two weeks), medium-run (IMFs 3-4: above two weeks and less than 12 weeks) and long-run (IMF5: above 12 weeks). Table 1 reports some measures which are given to assess IMFs: mean period of each IMF, correlation between each IMF and the original data series and the variance percentage of each IMF. The mean period corresponds to the value derived by dividing the total number of points by the number of peaks for each IMF. The Pearson and Kendall Rank coefficients are utilized to determine the correlation among IMFs and the original data. Because IMFs are intrinsically independent, it is possible to sum up the variances and employ the percentage of variance to measure the contribution of each IMF to the original data. The variables of interest (BP and SPI) are driven by the same components. In particular, both the U.S. stock price index and the Bitcoin price are explained by short-term hidden factors (IMF1and IMF2). To test if Bitcoin joins precious metals in safe haven properties in the aftermath of 2016 U.S presidential elections outcomes, we have estimated the multi-scale correlation between Bitcoin price and S&P500 stock price index, even if we control for gold and silver prices. We define a safe haven depending on the sign of the coefficients associated to the gold price. If the coefficient is negative or insignificant (i.e., uncorrelated), then Bitcoin, gold or silver may be served as a hedge or a safe haven. We include also WTI as a potential fundamental of stock price valuation. Table 2 reports the OLS-based EEMD regression' results.

During the post-2016 U.S. election period, Bitcoin price and S&P 500 index are likely to be negatively correlated in the short and the long term, but uncorrelated in the medium-run. This result underscores a new confidence in Bitcoin as a safe haven. As is with every other political issue regarding Trump's administration, it remains unclear what to expect. But there appears to be a quite general consensus in the Bitcoin community that whatever Trump's policies turn out to be, Bitcoin will benefit largely. This may explain the supported Bitcoin safe haven property. The reason that is making Bitcoiners hopeful about Trump's triumph is the inclusion of Bitcoin supporters like Peter Thiel, Balaji Srinivasan and Mick Mulvaney in his team. Peter Thiel is a technology entrepreneur and investor; he is the co-founder of PayPal, a Bitcoin enthusiast and has invested into multiple Bitcoin companies. Balaji Srinivasan is one of the best-funded Bitcoin startups so far. He is the co-founder and CEO of 21; the latter has developed a full stack set of technologies for practical Bitcoin micropayments. Also, Mick Mulvaney, the designated Director of the Office of Management and Budget under Trump's presidency, is viewed as one of the most representatives of the crypto community since he is more outspoken about Blockchain technology and Bitcoin. Having Bitcoin believers in the Trump's team is a win for the Bitcoin community in the whole. More interestingly, Donald Trump's election victory has sent U.S markets on a tumultuous ride. Markets are reacting as investors find out how heavy are the president-elect's statements on trade, fiscal policy and regulation. And as experienced after Trump's win, the uncertainty can encourage people to hoard assets such as Bitcoin and precious metals, both perceived as a hedge against uncertainty. But considering the sizable market volatility and the changes in traders' attitudes, it's natural to wonder whether Bitcoin can be viewed as a purely safe haven, which may explain the decreasing correlation between Bitcoin price and U.S stock price over time.

Unlike Bitcoin, the precious metals (gold and with less extent silver) exert a non-negative influence on the U.S. stock price index in the short and medium-run (IMF1, IMF2 and IMF3) and negative effect in the long term (IMF4 and IMF5). This means that the property of gold as well as silver is time-varying. In particular, they act as weak safe haven for S&P500 price index in short and intermediate time scales, and as a hedge in the longer time horizons. Typically, when the economy witnessed an evolving volatility that may impede stocks' valuation, investors may shift their funds from stocks and invest them in the gold and silver markets until the economy rebounds. In this context, precious metals could act as a stabilizer control in investment portfolios, and play as safe haven during turbulent times [START_REF] Baur | Is gold a Hedge or Safe Haven? An Analysis of Stocks Bonds and Gold[END_REF]. Also, traders tend to go into more liquid assets when market turbulence emerges and uncertainty increases. As an extremely liquid asset even in periods of market turmoil, gold can be served as a hedge.

Furthermore, there are swings from positive to negative correlation between WTI and S&P 500 over time, reflecting the volatile and the speculative behavior of World oil market. This finding seems against the alternative of adding crude oil to serve as a commodity diversifier in uncertain context.

Robustness

There exist different ways to check the robustness of our results. We carried out a series of robustness checks. First, we re-examine the correlation while replacing the nominal stock price by the real stock prices4 . While stock prices are heavily determined by financial variables [START_REF] Valcarcel | The dynamic adjustments of stock prices to inflation disturbances[END_REF], the prominence of macroeconomic variables cannot be ruled out [START_REF] Goyal | A Comprehensive Look at the Empirical Performance of Equity Premium Prediction[END_REF][START_REF] Rapach | Forecasting Stock Returns. Handbook of Economic Forecasting[END_REF]. Inflation is considered as one of the most potential macroeconomic variables believed to be related to stock prices. Even though inflationary shocks may have weak long-run impact on stock returns owing to monetary non-neutrality, it was largely documented that stock prices can be influenced by inflation in the short-run [START_REF] Valcarcel | The dynamic adjustments of stock prices to inflation disturbances[END_REF]Bjørnland and Jacobsen 2013). This underscores the importance to replace the nominal stock price by the real stock prices while adjusting the stock market pricing for inflation. This task may allow us to address whether the U.S monetary policy can stimulate the performance of stock market over the great uncertainty surrounding the U.S presidential elections. The results are reported in Table 3. Using another U.S. stock price proxy, we confirm that:

(i)

The hedge and safe-haven properties of Bitcoin, gold and silver vary depending to different time-horizons.

(ii)

Bitcoin poorly acts as a safe haven in the short and the long-run.

(iii) The gold and with less extent silver appear as interesting assets to hedge against unexpected events and uncertainty in the short-term. But they lost their hedging and safe haven properties in the long-run. (iv) The viability of oil as diversifier for a portfolio that is formed by U.S stocks remains unproven. Second, we replace the spot prices of gold and silver by their future prices. It must be stressed at this stage that the futures prices of precious metals are cleared via a centralized exchange with standardized contracts and complete fiduciary transparency. With heavily regulated markets, investors have access nearly 24hrs by day. On the contrary, the spot prices of the metals under study are completely unregulated. In particular, spot gold and silver dealers undertake their own prices and policies. In this way, these prices do not often reflect the world gold and prices.

The spot and futures prices are closely related via arbitrage strategies. For financial assets, if all arbitrage is possible, the forward price F t t+1 in t for t+1, is equal to the spot price S t at date t and the interest cost foregone over the period (r t S t ). F t t+1 = S t + r t S t (5) Since the storage costs must be accounted for [START_REF] Fama | Business cycles and the behavior of metal prices[END_REF], we followed [START_REF] Coudert | Gold and financial assets: are there any safe havens in bear markets?[END_REF] by including the cost of physical storage w t and by deducting then the convenience yield c t5 to the forward price yet defined in Equation ( 5). F t t+1 = S t + r t S t + w t -c t (6)

Using two different alternative prices of gold and silver, we robustly show that:

(i) Bitcoin may serve as a weak safe haven in the short-run and the longrun (negative correlation), and as a hedge in the medium term (uncorrelated with U.S. stock price). (ii) Precious metals lost their safe haven properties as their correlation with U.S. stock price index are declining from one frequency to another. (iii) Including crude oil in a portfolio formed by U.S stock price to serve as a commodity diversifier in turbulent times remains unsupported.

Discussion and conclusion

Donald Trump is one of the most controversial figures of the current times. So much so, that his victory in the U.S. presidential elections had led to a great uncertainty in the world economy. In the aftermath of Trump's win, a particular interest was devoted to digital assets (in particular, Bitcoin), strengthening its status as the modern safe haven. As safe havens and hedge rely on co-movements between assets in times of market turmoil, this paper seeks to examine the timevarying relationship between Bitcoin price and U.S. stock price index during the post-U.S. election period. We then compare Bitcoin hedging and safe haven properties to the traditional assets (in particular, gold, silver).

An effective analysis of such complex issue needs the exclusion of the parametric approaches that use predetermined patterns. The core focus is to suggest a newly and adequate econometric tool for reading data and permitting the investigation with respect to different frequencies, namely Ensemble Empirical Mode Decomposition. Indeed, the price of several assets (Bitcoin, U.S. stocks, gold, silver and oil) data were decomposed into several intrinsic mode functions, even if we control for aliasing in the time-frequency distribution.

Our results indicate that the hedge and safe-haven properties of Bitcoin, gold and silver for U.S. stock market is not constant over time. Bitcoin acts as a weak safe haven in the short-run, and as a hedge in the medium-and the long-run. We show also that gold and silver lost their hedge and safe haven properties as the comovement among precious metals and U.S. stock price index decreases over time. Investors and traders are generally interested in hedges that mitigate the volatility of their portfolio, but also they are likely interested in buying some sort of insurance against extreme tail events such as the uncertainty surrounding the unanticipated U.S. election outcome. Bitcoin has several properties that make it a very interesting asset in both cases. Currently, the loss of faith in the stability of banking system and future economic security worsened, and market uncertainty heightened across the globe. But Bitcoin which lives outside the confines of a single country's politics has profited from the current ongoing volatility. While these properties may justify that Bitcoin serves as a hedge for U.S. stock price index in turbulent times, they explain also why Bitcoin cannot be considered as a strong safe haven. When market turmoil arises and uncertainty rises, investors are known to sell -risky‖ assets and buy -safer‖ assets, also known as -flights to safety‖ [START_REF] Baele | ‗Flights to Safety[END_REF]. And Bitcoin does not hold this property. Being a cryptocurrency, Bitcoin is highly sensitive to cyber-attacks which may destabilize its whole system (Barber et al. 2012).

Further, Bitcoin and precious metals do not act in similar way for U.S. stock market. From a legal perspective, Bitcoin does not appear to share the features of traditional safe-haven investments. Specifically, the validity of Bitcoin as a hedge and a safe haven may encounter a number of obstacles. Although Bitcoin is a liquid asset even in times of market upheaval, it is a high-risk, volatile and speculative investment. However, as extremely liquid assets even in turbulent times, gold and silver can be characterized as safer assets.

Figure 1 .

 1 Figure 1. The EMD decomposition of a simple signal

Figure 3

 3 Figure 3 confirms the previous results showing that the high frequency components are the major contributors of BP and SPI.

Figure 3 .

 3 Figure 3. The hidden characteristics of Bitcoin price and U.S. stock price index BP

Table 1 . The IMFs features

 1

		Mean	Pearson	Kendall	variance as % of
		period	correlation	correlation	the sum of IMFs
			BP
	IMF1	3.62	0.262***	0.199**	44.98%
	IMF2	7.94	0.256*	0.234**	29.26%
	IMF3	8.39	0.195*** 0.167***	9.23%
	IMF4	11.93	0.131**	0.116**	7.14%
	IMF5	25.04	0.112**	0.099*	6.68%
			SPI
	IMF1	4.32	0.183*	0.175**	36.98%
	IMF2	7.69	0.277**	0.259**	41.12%
	IMF3	8.16	0.113*	0.089**	1.18%
	IMF4	13.18	0.218*** 0.204***	3.46%
	IMF5	38.25	0.096**	0.075***	17.27%

Notes: *, **, ***: Correlations are significant at the levels of 0.01, 0.05 and 0.1.

Table 2 . Multi-scale estimates of the relationship between Bitcoin price and U.S. stock price

 2

		IMF1	IMF2	IMF3	IMF4	IMF5
			Dependent variable: SPI
	C	-0.5408*	0.9785**	0.663966	0.796386*	1.108502
		(0.0439)	(0.0041)	(0.1700)	(0.0304)	(0.1989)
	SPI(-1) -0.27057*	-1.059**	-0.32362*	0.255759*	0.235810
		(0.0426)	(0.0071)	(0.0345)	(0.0855)	(0.6221)
	BP	-0.141**	-0.129**	0.543518	-0.14424	-0.1102*
		(0.0007)	(0.0095)	(0.4610)	(0.2131)	(0.3202)
	gold	0.05799*	0.0417*	0.0278*	-0.0936**	-0.0931*
		(0.0153)	(0.0173)	(0.0306)	(0.0016)	(0.0460)
	silver	0.0446**	0.0221*	0.012213*	-0.0036**	-0.0161*
		(0.0050)	(0.0235)	(0.0164)	(0.0036)	(0.0761)
	WTI	-0.09861*	0.1158*	0.09687*	-0.1410**	0.0984*
		(0.0865)	(0.0137)	(0.0672)	(0.0059)	(0.0126)
	R2	0.85	0.83	0.84	0.86	0.85

Notes: (.): p-values. ***, ** and * in the table denote statistical significant coefficients at 1 per cent, 5 per cent and 10 per cent level, respectively.

Table 3 . Multi-scale estimates of the relationship between Bitcoin price and real U.S. stock price

 3

		IMF1	IMF2	IMF3	IMF4	IMF5
			Dependent variable: real SPI
	C	0.565629	0.891338*	-0.17745*	-0.0981**	-0.50874*
		(0.9331)	(0.0315)	(0.0597)	(0.0079)	(0.0719
	SPI(-1) -0.088349	0.054829	-0.01213*	0.1181*	0.125603
		(0.8403)	(0.2674)	(0.0538)	(0.0556)	(0.7987
	BP	-0.14348**	-0.15141*	-0.08592	-0.1433*	-0.12482*
		(0.0095)	(0.0636)	(0.3490)	(0.0116)	(0.0170)
	gold	0.094893*	0.05097**	-0.108265	-0.135**	0.092123
		(0.0216)	(0.0042)	(0.3479)	(0.0058)	(0.1632)
	silver	0.049722*	0.028905*	0.02015**	-0.0863*	-0.0663**
		(0.0343)	(0.0474)	(0.0091)	(0.0486)	(0.0054)
	WTI	-0.10139*	0.0529**	-0.1235**	-0.1164*	0.02374**
		(0.0835)	(0.0016)	(0.0021)	(0.0111)	(0.0069)
	R2	0.90	0.84	0.89	0.90	0.89
	Notes: (.): p-values. ***, ** and * in the table denote statistical significant coefficients at 1 per cent, 5
	per cent and 10 per cent level, respectively.

Table 4 . Multi-scale estimates of the relationship between Bitcoin price and U.S. stock price by replacing the spot prices of gold and silver by the forward prices

 4

		IMF1	IMF2	IMF3	IMF4	IMF5
			Dependent variable: SPI
	C	-0.40923*	0.521058	-0.4009*	-0.654***	0.21386**
		(0.0352)	(0.3894)	(0.0305	(0.0000)	(0.0037)
	SPI(-1)	-0.204381	0.045861	-0.21213*	0.19864*	0.28438**
		(0.9210)	(0.1527)	(0.0326)	(0.0425)	(0.0045)
	BP	-0.148***	-0.12351*	0.1939	-0.096***	-0.1114**
		(0.0000)	(0.0950)	(0.2385)	(0.0000)	(0.0095)
	gold	0.0746***	0.13451	0.45216	-0.0613**	-0.0341***
		(0.0002)	(0.1843)	(0.5133)	(0.0045)	(0.0001)
	silver	0.0446**	0.0221*	0.01218	-0.0356*	-0.0161*
		(0.0050)	(0.0235)	(0.3164)	(0.0145)	(0.0761)
	WTI	-0.1056**	0.08627*	-0.1478*	-0.119***	0.09653***
		(0.0031)	(0.0290)	(0.0997)	(0.0000)	(0.0001)
	R2	0.85	0.83	0.84	0.86	0.85

Notes: (.): p-values. ***, ** and * in the table denote statistical significant coefficients at 1 per cent, 5 per cent and 10 per cent level, respectively.

For detailed discussion of the EMD technique and comparison to other time series analysis tools, you can refer to[START_REF] Huang | The empirical mode decomposition method and the Hilbert spectrum for non-stationary time series analysis[END_REF] Flandrin et al. (2004).

Since inflation rate is unavailable in daily frequency, we use Data Frequency Conversion from low (i.e., inflation monthly data) to high (i.e., inflation daily data) frequency via Eviews 9. For more details about the conversion procedure, you can refer to this link: http://www.eviews.com/Learning/freqconv.html. Monthly inflation data are collected from EcontatsTM.

The convenience yield represents the gain obtained through storing the commodity, and that stems from the uncertainty surrounding the future production. For the future prices of gold and silver, the data are available in Datastream database.

Appendix

Bitcoin price and S&P 500 index: IMFs derived from EEMD BP