J. Math. Anal. Appl. 416 (2014) 735–747

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

d-Symmetric d-orthogonal polynomials of Brenke type

Youssef Ben Cheikh a, Neila Ben Romdhane b,∗

a Département de Mathématiques, Faculté des Sciences de Monastir, Université de Monastir, Tunisiab École Supérieure des Sciences et de Technologie de Hammam Sousse, Université de Sousse, Tunisia

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 June 2013Available online 26 February 2014Submitted by M.J. Schlosser

Keywords:Orthogonal polynomialsGenerating functionsd-Symmetric polynomialsd-Orthogonal polynomialsBrenke polynomialsd-Dimensional functional vectorComponent sets

In this paper, we characterize the d-symmetric d-orthogonal polynomials ofBrenke type. We obtain two new families of polynomials and the moments ofthe corresponding d-dimensional vectors of linear functionals. Weights, providingintegral representations for these moments, were also given.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction

Let P be the linear space of polynomials with complex coefficients. A polynomial sequence {Pn}n�0 in Pis a polynomial set (PS, for short) if and only if degPn = n for all non-negative integers n.

A PS {Pn}n�0 is called symmetric if it fulfills Pn(−x) = (−1)nPn(x), n � 0.Al-Salam [1] and, later on, von Bachhaus [27], by using different techniques, have characterized the

classical symmetric orthogonal polynomials (Hermite and Gegenbauer) as the only standard orthogonalpolynomials defined by the generating function: G(2xt− t2) =

∑∞n=0 cnPn(x)tn, cn �= 0, n � 0.

Chihara determined all orthogonal polynomial sequences of Brenke type [8]. In particular, the four citedsymmetric ones are: the Generalized Hermite’s polynomials, a particular case of the Stieltjes–Wigert polyno-mials, and the two sets of polynomials related to Al-Salam–Carlitz and Wall PSs. We note that the Brenketype standard orthogonal polynomial sequences are defined as follows [7]

A(t)B(xt) =∞∑

n=0Pn(x)tn,

* Corresponding author.

E-mail addresses: youssef.bencheikh@planet.tn (Y. Ben Cheikh), neila.benromdhane@ipeim.rnu.tn (N. Ben Romdhane).

http://dx.doi.org/10.1016/j.jmaa.2014.02.0460022-247X/© 2014 Elsevier Inc. All rights reserved.

736 Y. Ben Cheikh, N. Ben Romdhane / J. Math. Anal. Appl. 416 (2014) 735–747

where A and B are two formal power series satisfying:

A(t) =∞∑

n=0ant

n, B(t) =∞∑

n=0bnt

n, a0bn �= 0, ∀n ∈ N.

Extensions of symmetric and orthogonal polynomials are respectively d-symmetric and d-orthogonal poly-nomials, d being a positive integer number. These notions are defined as follows:

A PS {Pn}n�0 is called d-symmetric if it fulfills [20]

Pn(ωx) = ωnPn(x), ω = exp(2iπ/(d + 1)

). (1.1)

{Pn}n�0 is a d-orthogonal polynomial set (d-OPS, for short) if there exists a d-dimensional vector of linearfunctionals, U = (u0, u1, . . . , ud−1), such that [19,26]:{

〈uk, PmPn〉 = 0 if m > nd + k, n � 0,

〈uk, PnPnd+k〉 �= 0, n � 0, k ∈ {0, 1, . . . , d− 1},(1.2)

and where 〈u, P 〉 denotes the effect of the linear functional u on the polynomial P .Recently, we have generalized the above results by von Bachhaus to d-symmetric d-OPSs [4]. In fact, we

proved that there are exactly 2d d-symmetric classical d-OPSs and that these polynomials have a generatingfunction of the form G((d + 1)xt− td+1).

Our purpose in this work is to solve the following problem:

(P): Find all d-symmetric d-OPSs of Brenke type.

That is to say: Find all d-OPSs, {Pn}n�0, generated by

A(td+1)B(xt) =

∞∑n=0

Pn(x)tn, (1.3)

where A and B are two formal power series satisfying:

A(t) =∞∑

n=0ant

n, B(t) =∞∑

n=0bnt

n, a0bn �= 0, ∀n ∈ N. (1.4)

This problem was treated for the following two particular cases.

1. The case B(t) = exp(t) was considered by Douak [11]. He obtained that A(t) = exp(t), and the resultingpolynomials are an extension of the Hermite polynomials. They were called Hermite type d-OPS andthey are, in fact, the Gould–Hopper polynomials [15].

2. The case B(t) = Eμ(t) was treated by the first author and Gaied [6], where the function Eμ is definedby,

Eμ(x) =∞∑

n=0

xn

γμ(n) , γμ(0) = 1, γμ(n) = (n + 2μθn)γμ(n− 1), n � 1, θ2n = 0, θ2n+1 = 1. (1.5)

They obtained that A(t) = exp(t), and the polynomials were named the Gould–Hopper type polynomi-als.

Y. Ben Cheikh, N. Ben Romdhane / J. Math. Anal. Appl. 416 (2014) 735–747 737

Rajković and Marinković [22] investigated the case A(t) = 1/eqd+1(at) and B(t) = eq(t) where theq-exponential function is defined by (see for instance (10.2.9) in [3])

eq(t) =∞∑

n=0

tn

[n]q!,

where

[n]q = 1 − qn

1 − q, [n]q! = [n]q[n− 1]q · · · [1]q, [0]q! = 1.

They do not involve the notion of d-orthogonality. The obtained polynomials are a q-extension of the Hermitepolynomials. They were called q-analogies of generalized Hermite’s polynomials.

The outline of the paper is as follows. Following the introduction, Section 2 is devoted to the solution ofthe problem (P) which consists in keeping a recurrence relation satisfied by the coefficients (an) and (bn)(Eq. (1.4)) and its resolution. We obtain two new families of d-symmetric d-OPSs generated by functionsexpressed in terms of the hypergeometric function and the basic hypergeometric series. These results aregiven in Theorem 2.1. Well known examples of symmetric orthogonal polynomials and d-symmetric d-OPSsare particular cases of these families. They are summarized in Table 1 and Table 2. Finally, Section 3deals with some properties for the obtained polynomials: The inversion formulas, the moments of thecorresponding d-dimensional vector of linear functionals are obtained in Theorem 3.2 by using the inversionformulas. The corresponding vectors of weights, providing of integral representations for the moments, aregiven in Theorems 3.3–3.4. Explicit expressions for the components of d-symmetric d-OPSs of Brenke typeare also given in Theorem 3.5.

Before discussing the above problem, let us recall some definitions which we need below.The shifted factorials are defined by

(β)0 = 1, (β)n = β(β + 1) · · · (β + n− 1), n � 1.

The standard notion for hypergeometric function is [24]

pFq

(β1, β2, . . . , βp

γ1, γ2, . . . , γq;x

)=

∞∑n=0

(β1)n(β2)n · · · (βp)n(γ1)n(γ2)n · · · (γq)n

xn

n! . (1.6)

The q-shifted factorials are defined by

(β; q)0 = 1, (β; q)n =n∏

k=1

(1 − βq(k−1)), n � 1,

and the multiple q-shifted factorials are

(β1, β2, . . . , βk; q)n =k∏

j=1(βj ; q)n.

The basic hypergeometric series rΦs (or q-hypergeometric series) is defined by [14]

rΦs

(β1, β2, . . . , βr

γ1, γ2, . . . , γs; q, x

)=

∞∑n=0

(β1, . . . , βr; q)n(γ1, . . . , γs; q)n

((−1)nq(n2))1+s−rxn

(q; q)n, (1.7)

where γ1, γ2, . . . , γs �= 1, q−1, q−2, . . . .Throughout the paper, we take the convention that an empty product is equal to 1 and an empty sum to 0.

738 Y. Ben Cheikh, N. Ben Romdhane / J. Math. Anal. Appl. 416 (2014) 735–747

2. d-Symmetric d-OPSs of Brenke type

The solution of the problem (P) is given in the following theorem.

Theorem 2.1. The d-symmetric d-OPSs of Brenke type are one of the following two sets:

1. The polynomials {P1,n}n�0 generated by

exp(atd+1)B1(xt) =

∞∑n=0

P1,n(x)tn, a �= 0, (2.1)

where

B1(t) =d∑

l=0

tl∏lk=1 vkmk

0Fd

(−m1 + 1, . . . ,ml + 1,ml+1, . . . ,md

; td+1∏dk=0 vk

),

where (vl)0�l�d are non-zero constants and ml �= 0,−1,−2, . . . , 1 � l � d.2. The polynomials {P2,n}n�0 generated by

eq(atd+1)B2(xt) =

∞∑n=0

P2,n(x)tn, a �= 0, (2.2)

where

B2(t) =d∑

l=0

tl∏lk=1 uk(ηk − 1)

0Φd

(−η1q, . . . , ηlq, ηl+1, . . . , ηd

; q, ql+1td+1∏dk=0 uk

),

where (ηl)1�l�d and (ul)0�l�d are such that ul �= 0 and ηl �= 1, q−1, q−2, . . . .

We note that the hypergeometric function obtained here is the hyper-Bessel function [10], and we remarkthat for ηl = qαl , 1 � l � d, αl �= 0,−1,−2, . . . , the first case is a limit case of the second one when q −→ 1.We have: limq �−→1 P2,n((q − 1)x) = P1,n(x). In fact, [17]

limq �−→1

eq(x) = exp(x) and limq �−→1 0Φd

(−qα1 , . . . , qαd

; q, (q − 1)d+1x

)= 0Fd

(−α1, . . . , αd

;x).

The proof is divided into two parts. First, we determine necessary and sufficient conditions for Brenke PSsto be d-symmetric d-orthogonal. We obtain a second-order recurrence relation satisfied by the coefficients(an) and (bn) (Eq. (2.3)). Then, in Section 2.2, we determine the solutions of the problem (P). Particularcases of these polynomials are given in Section 2.3.

2.1. Necessary and sufficient conditions

Without loss of generality, we assume that a0 = b0 = 1 and we put a1 = a.

Proposition 2.2. {Pn}n�0 is a d-symmetric d-OPS of Brenke type if and only if the coefficients of the formalseries in its generating function (1.3) satisfy

ak(rn − rn−k(d+1)) = aak−1(rn − rn−(d+1)), d + 1 � n, 0 � k �[

n], (2.3)

d + 1

Y. Ben Cheikh, N. Ben Romdhane / J. Math. Anal. Appl. 416 (2014) 735–747 739

where rn = bn−1/bn, r0 = 0, and with the regularity conditions:

rn �= 0, rn �= rn−(d+1), n � d + 1. (2.4)

To prove this result, we recall the following

Lemma 2.3. (See [12].) Let {Pn}n�0 be a monic d-OPS. Then, the following statements are equivalent:

1. {Pn}n�0 is d-symmetric.2. {Pn}n�0 satisfies a (d + 1)-order recurrence relation of the form{

Pn+d(x) = xPn+d−1(x) − CnPn−1(x), n � 1, Cn �= 0,

Pn(x) = xn, 0 � n � d.(2.5)

Proof of Proposition 2.2. Let {Pn}n�0 be a PS generated by (1.3). Multiplying the sums of Eq. (1.3) andusing the following formula [21]

∞∑n=0

∞∑k=0

A(k, n) =∞∑

n=0

[n/m]∑k=0

A(k, n−mk), (2.6)

we obtain Pn(x) =∑[n/d+1]

k=0 akbn−k(d+1)xn−k(d+1).

Let Pn(x) = Pn(x)bn

be the monic polynomial corresponding to Pn(x).By virtue of Lemma 2.3, {Pn}n�0 is a d-symmetric d-OPS if and only if

Pn+d = xPn+d−1 − CnPn−1, Cn �= 0, n � 1. (2.7)

If both sides of (2.7) are differentiated n + d− k(d + 1) times and x is equal to 0, we obtain:

ak−1Cn = ak

(bn−k(d+1)+d−1

bn+d−1−

bn−k(d+1)+d

bn+d

)bn−1

bn−k(d+1)+d, 1 � k �

[n + d

d + 1

]. (2.8)

Writing rn = bn−1bn

, then Eq. (2.8) takes the form

ak−1Cn = −akrnrn+1 · · · rn+d−1(rn+d − rn−(k−1)(d+1)−1). (2.9)

Taking k = 1 in (2.9), we obtain:

Cn = −arn · · · rn+d−1(rn+d − rn−1), n � 1, a �= 0. (2.10)

Replacing Cn by its expression in (2.9), we obtain the desired result. �2.2. Determination of the solutions

Now, we give the proof of our result.

Proof of Theorem 2.1. For k = 2 in Eq. (2.3), we obtain the necessary condition for n � 2(d + 1)(a2 − a2)rn + a2rn−(d+1) − a2rn−2(d−1) = 0. (2.11)

740 Y. Ben Cheikh, N. Ben Romdhane / J. Math. Anal. Appl. 416 (2014) 735–747

Let Δn,l = rn(d+1)+l. Then, it satisfies(a2 − a2)Δn,l + a2Δn−1,l − a2Δn−2,l = 0, 2 � n.

The solutions of this second-order recurrence relation are

• If a2 = a2/2: rn(d+1)+l = sl + nvl. (2.12)

• If a2 �= a2/2: rn(d+1)+l = hl − ul

(a2

a2 − a2

)n

. (2.13)

Taking n = 0 in (2.12), we deduce that: s0 = 0, sl = rl �= 0, 1 � l � d, and for n = 1, we have from theregularity conditions (2.4) vl = rd+1+l − rl �= 0, 0 � l � d.

Similarly, we prove in the second case, that rl = hl − ul �= 0, 1 � l � d, and ul �= 0.Next, we deal with each case separately and we determine the expressions of A(t) and B(t).

(1) Case a2 = a2/2: (i) Determination of A1(t): Substituting Eq. (2.12) into Eq. (2.3), we get kakvl =aak−1vl. Since vl �= 0, we obtain (kak − aak−1) = 0, ∀k � 1, and then

a1,n := an = an

n! , n � 0.

Replacing a1,n by its expression in A1(t) =∑∞

n=0 a1,ntn, we obtain, A1(t) = exp(at).

(ii) Determination of B1(t): We have

rn(d+1)+l = vl(ml + n), 0 � n, 0 � l � d, (2.14)

with m0 = 0 and ml = sl/vl, vl �= 0.Now, we proceed to determine the coefficient bn. Since rn = bn−1

bn, b0 = 1, we obtain

bn(d+1)+l = 1rn(d+1)+l · · · rn(d+1) · · · r(n−1)(d+1)+d · · · r(n−1)(d+1) · · · rd · · · r1

. (2.15)

Replacing rn by its expression (2.14), we get

bn(d+1)+l = 1vl(ml + n) · · · v0(m0 + n)

× 1vd(md + n− 1) · · · v0(m0 + n− 1)

...

× 1vd(md) · · · v1(m1)

,

which can be written as follows

b1,n(d+1)+l := bn(d+1)+l = 1∏lk=1 vkmk

∏dk=0 v

nk

∏lk=1(mk + 1)n

∏dk=l+1(mk)nn!

. (2.16)

Replacing b1,n by its expression in B1(t) =∑d

l=0∑∞

n=0 b1,n(d+1)+ltn(d+1)+l, we obtain the desired re-

sult (2.1).

(2) Case a2 �= a2/2: (i) Determination of A2(t): Substituting (2.13) into (2.3), we get an = a 1−q1−qk

an−1,n � 1, q = a2−a2

a2. Thus a2,n := an = an

[n]q! , n � 0.Replacing a2,n by its expression in A2(t) =

∑∞a2,nt

n, we obtain A2(t) = eq(at).

n=0

Y. Ben Cheikh, N. Ben Romdhane / J. Math. Anal. Appl. 416 (2014) 735–747 741

Table 1Particular cases of polynomial sets for d = 1.

Polynomial sets, {Pn}n�0 B1(t) B2(t)m; v0; v1 q; η;u1;u0

Generalized Hermite’s polynomials [25,9]exp(−at2)Eμ(xt) =

∑∞n=0 Hμ

n(x) tn

γμ(n)

2μ+12 ; 4; 1 –

Symmetric Wall polynomials [9]A (t2)[B(x2t2; b, q) + xtB(x2t2; bq, q)] =

∑∞n=0

Yn(x)tncn

– q−1; b−1

b1−b ; 1 − b

Al-Salam Carlitz polynomials [2]eq(xt)eq−2 (− qt2

1+q2 ) =∑∞

n=0 U (−1)n (x) tn

(q;q)n

– q−2; q−1q

1−q ; 11−q

Stieltejes–Wigert polynomials [8,9]

A (t2)[B(x2t2; p, q)+ q1/4xt1−p B(qx2t2; pq, q)] =

∑∞n=0

qn2/4

cnTn(x; p, q)tn

– q; p−q−1/4;−q1/4

(ii) Determination of B2(t): Writing rn as follows

rn(d+1)+l = −ulq−n

(1 − ηlq

n),

where ηl = hl

uland replacing rn by its expression in (2.15), we get

b2,n(d+1)+l := bn(d+1)+l = (−1)l{(−1)nq(n2)}(d+1)qn(l+1)

ul(1 − ηlqn) · · ·u0(1 − η0qn)

× 1ud(1 − ηdqn−1) · · ·u0(1 − η0qn−1)

...

× 1ud(1 − ηd) · · ·u1(1 − η1)

.

Since (θ; q)n+1 = (1 − θ)(θq; q)n and η0 = 1, we can then write b2,n as follows

b2,n(d+1)+l = {(−1)nq(n2)}(d+1)qn(l+1)∏l

k=1 uk(ηk − 1)∏d

k=0 unk

∏lk=1(ηkq; q)n

∏dk=l+1(ηk; q)n(q; q)n

. (2.17)

Substituting b2,n in B2(t) =∑d

l=0∑∞

n=0 b2,n(d+1)+ltn(d+1)+l, we obtain the desired result (2.2). �

2.3. Particular cases

In this subsection, we show that well known examples of polynomials in the literature, are particular casesof Theorem 2.1. In fact, for d = 1, the particular cases of these polynomials are summarized in Table 1. Wegive for B1(t) the parameters m, v0, v1 and for B2(t) the parameters q, η, u0, u1.

For d � 1, the particular cases of these polynomials are summarized in Table 2. We give for B1(t) theparameters vl, l = 0, . . . , d, and ml, l = 1, . . . , d, and for B2(t) the parameters q, ul, l = 0, . . . , d, and ηl,l = 1, . . . , d.

3. Properties

In this section, for d-symmetric d-OPSs of Brenke type obtained in Theorem 2.1, we determine theinversion formulas, the two d-dimensional functional vectors for which the d-orthogonality holds and thecomponent sets.

742 Y. Ben Cheikh, N. Ben Romdhane / J. Math. Anal. Appl. 416 (2014) 735–747

Table 2Particular cases of polynomial sets for d � 1.

Polynomial sets, {Pn}n�0 B1(t) B2(t)vl, l = 0, . . . , d q, ul, l = 0, . . . , dml, l = 1, . . . , d ηl, l = 1, . . . , d

Gould–Hopper polynomials [15,11]exp(xt − td+1

d!(d+1)2 ) =∑∞

n=0 Hn(x, d)tnvl = d + 1 –ml = l

d+1

Gould–Hopper type polynomials [6]

exp(atd+1)Eμ(xt) =∑∞

n=0 P (d+1)n (x; a, μ) tn

γμ(n)

vl = (d+1)(l+2μθl)l+1+2μθl+1 –

ml = l+1+2μθl+1d+1

q-Analogues of generalized Hermite’s polynomials [22]eq(xt)

eqd+1 (atd+1) =

∑∞n=0 hn,d+1(x; q) tn

[n]q!–

q′ = q−(d+1)

ul = ql

1−q

ηl = q−l

3.1. Inversion formulas

Theorem 3.1. For the PSs {P1,n}n�0 and {P2,n}n�0 generated respectively by (2.1) and (2.2) the followinginversion formulas hold:

xp(d+1)+l = p!d∏

k=0

vpk

l∏k=1

vkmk(mk + 1)pd∏

k=l+1

(mk)pp∑

k=0

(−a)p−k

(p− k)! P1,k(d+1)+l(x). (3.1)

xp(d+1)+l =∏d

k=0 upk

∏lk=1 uk(ηk − 1)

(−1)p(d+1)q(l+1)p+(p2)(d+1)

(η1q, . . . , ηlq, ηl+1, . . . , ηd, q; q)pp∑

k=0

(−a)p−k

[p− k]q−1 !P2,k(d+1)+l(x). (3.2)

Proof. From the generating function of the polynomials {P1,n}n�0, we have

B1(xt) =∞∑

n=0P1,n(x)tn

∞∑k=0

(−a)ktk(d+1)

k! .

Using the formula (2.6), we obtain

B1(xt) =∞∑

n=0

[ nd+1 ]∑k=0

(−a)k

k! P1,n−(d+1)k(x)tn,

writing n = p(d + 1) + l, then

B1(xt) =d∑

l=0

∞∑p=0

p∑k=0

(−a)k

k! P1,(p−k)(d+1)+l(x)tp(d+1)+l. (3.3)

Replacing 0Fd(xt) with its definition expression in B1(xt), we get

B1(xt) =d∑

l=0

∞∑p=0

(xt)p(d+1)+l

p!∏d

k=0 vpk

∏lk=1 vkmk(mk + 1)p

∏dk=l+1(mk)p

. (3.4)

Finally, by comparing the two expressions of B1(xt), (3.3) and (3.4), we deduce the first result.For the second result, we proceed similarly using the property 1/eq(t) = eq−1(−t). �These results generalize the known inversion formulas of the polynomials particular cases of Theorem 2.1.

Y. Ben Cheikh, N. Ben Romdhane / J. Math. Anal. Appl. 416 (2014) 735–747 743

3.2. d-Dimensional functional vectors

Our purpose in the sequel is to determine the d-dimensional linear functional vectors for which thed-orthogonality of the resulting d-OPSs in Theorem 2.1 holds, then, we express the moments by means ofintegral representations. First, we recall the following.

Let {Pn}n�0 be a monic PS in P. The corresponding dual sequence, {Lr}r�0, is defined by

〈Lr, Pn〉 = δr,n, r � 0, n � 0,

where δr,n designates the Kronecker symbol.We denote by (Lr)n := 〈Lr, x

n〉 the moments of Lr.Maroni [19] proved that if {Pn}n�0 is d-orthogonal then it is d-orthogonal with respect to Λ =

(L0, . . . , Ld−1) the d first linear functionals of the dual sequence.Starting from the inversion formula xn =

∑nj=0 Ij(n)Pj(x), the moments of the d-dimensional functional

vector, Λ = (L0, . . . , Ld−1), related to {Pn}n�0 are expressed as follows

⟨Lr, x

n⟩

={Ir(n) if n � r,

0 otherwise.

For the polynomials obtained in Theorem 2.1, we have

Theorem 3.2. The polynomials {Pi,n}n�0, i = 1, 2, are d-OPSs with respect to the d-dimensional functionalvectors Λi = (Li

r)0�r�d−1 given by the following moments

(L1r

)p(d+1)+l

= δr,l(−a)pd∏

i=0vpi

l∏i=1

(mi + 1)pd∏

i=l+1

(mi)p. (3.5)

(L2r

)p(d+1)+l

= δr,l(−1)pdap(1 − q)p

∏di=0 u

pi

q(p(l+1)+(p2)d)

(η1q, . . . , ηlq, ηl+1, . . . , ηd; q)p. (3.6)

Proof. Eq. (3.1) can be written as follows

xp(d+1)+l =p(d+1)+l∑

j=0Ij(p(d + 1) + l

)P1,j(x)

where P1,n = 1b1,n

P1,n, b1,n is given by (2.16) and

Ij(p(d + 1) + l

)=

{p!(−a)p−kb1,j

(p−k)!∏d

i=0 vpi

∏li=1 vimi(mi + 1)p

∏di=l+1(mi)p, if j = k(d + 1) + l,

0, otherwise.

Since, for j = r, we have k = 0, then we obtain the desired result (3.5).The same proof for (3.6). �Next, we give the d-dimensional vectors of weight functions for (Li

r)0�r�d−1, i = 1, 2.

Theorem 3.3. For the polynomials {P1,n}n�0, the moments given by (3.5), with mi > 0, 1 � i � d, anda∏d

i=0 vi < 0, have the following integral representations.

⟨L1r, x

n⟩

= δr,l

∞∫unψr(u) du, n = k(d + 1) + l (3.7)

0

744 Y. Ben Cheikh, N. Ben Romdhane / J. Math. Anal. Appl. 416 (2014) 735–747

where

ψr(u) = (d + 1)u−(r+1)∏di=1 Γ (γi)

Gd,00,d

(ud+1

−a∏d

k=0 vk

∣∣∣−γ1, . . . , γd

),

Γ is the Gamma function, γi = mi +1, 1 � i � r, γi = mi, r+1 � i � d, and Gm,np,q designates the Meijer’s

G-function (see, for instance, [18, p. 143]).

Proof. Using the fact that (β)n = Γ (β + n)/Γ (β), the moments of the polynomials {P1,n}n�0, Eq. (3.5),can be written as

⟨L1r, x

k(d+1)+l⟩

= δr,l(−a)kd∏

i=0vki

∏ri=1 Γ (mi + 1 + k)

∏di=r+1 Γ (mi + k)∏r

i=1 Γ (mi + 1)∏d

i=r+1 Γ (mi). (3.8)

The formula [18, p. 157, Case 2]

∞∫0

ys−1Gm,np,q

(ηy

∣∣∣ βp

γq

)dy = η−s

∏mi=1 Γ (γi + s)

∏ni=1 Γ (1 − βi − s)∏q

i=m+1 Γ (1 − γi − s)∏p

i=n+1 Γ (βi + s)

is valid, in our case for s = k, m = q = d, n = p = 0, η = 1/(−a∏d

i=0 vi) > 0, then, we obtain

⟨L1r, x

k(d+1)+l⟩

= δr,l∏di=1 Γ (γi)

∞∫0

yk−1Gd,00,d

(y

−a∏d

i=0 vi

∣∣∣−γ1, . . . , γd

)dy. (3.9)

By means of the change of variables y = ud+1, we get the desired result. �To state the second theorem, we recall the following:The q-Jackson integration is defined by [16]

∞∫0

f(x) dq(x) = (1 − q)n=+∞∑n=−∞

f(qn

)qn.

The G function, related to the q-gamma function Γq, is defined by

G(qα

)= 1∏∞

n=0(1 − qα+n)= (1 − q)α−1∏∞

n=0(1 − q1+n)Γq(α), α �= 0,−1,−2, . . . .

The basic Meijer’s G-function Gm,nA,B (for more details, see, for instance, [23])

Gm,nA,B

[x; q

∣∣∣ a1, . . . , aAb1, . . . , bB

]= 1

2iπ

∫C

∏mj=1 G(qbj−s)

∏nj=1 G(q1−aj+s)∏B

j=1 G(q1−bj+s)∏A

j=1 G(qaj−s)G(q1−s)πxs

sin πsds.

Theorem 3.4. For the polynomials {P2,n}n�0, the moments given by (3.6), with a∏d

i=0 ui < 0 and ηi = qαi ,αi �= 0,−1,−2, . . . , 1 � i � d, have the following integral representations.

⟨L2r, x

n⟩

= δr,l

∞∫unϕr(u) d(q′)1/(d+1)u, n = k(d + 1) + l (3.10)

0

Y. Ben Cheikh, N. Ben Romdhane / J. Math. Anal. Appl. 416 (2014) 735–747 745

where

ϕr(u) = log(q′)η1/2

π((q′)1/(d+1) − 1)u−( d−1

2 +l)∏di=1 G((q′)bi+ 1

2 )Gd+1,0

0,d+1

[ηud+1; q′

∣∣∣−b1, . . . , bd+1

],

q′ = q−1, bi = αi + 1/2, 1 � i � r, bi = αi − 1/2, r + 1 � i � d, bd+1 = 1, and η−1 = a(1 −q′)(q′)−

∑di=1 αi

∏di=0 ui.

Proof. Using the fact that [17] (qα; q)k = (−1)kqαk+(k2)(q−α; q−1)k, and (qα; q)k = G(qα+k)

G(qα) , we get (qα; q)k =

(−1)kqαk+(k2)G((q′)α+k)G((q′)α) .

Then, the moments of the polynomials {P2,n}n�0, Eq. (3.6), can be written as

⟨L2r, x

k(d+1)+l⟩

= δr,l

(a(q′ − 1

)(q′)−∑d

i=1 αi

d∏i=0

ui

)k∏di=1 G((q′)bi+1/2+k)∏di=1 G((q′)bi+1/2)

. (3.11)

On the other hand, from the definition of the basic Meijer’s G-function Gd+1,00,d+1 and the q-Mellin inversion

formula [13], we have:

∞∫0

ys−1Gd+1,00,d+1

[ηy; q′

∣∣∣ −b1, . . . , bd+1

]dq′y = π(q′ − 1)η−s

log(q′) sin(πs)

d∏i=1

G((q′)bi+s)

.

For s = k + 1/2, and by using the q-analogue of the integration theorem by the change of variable y =ud+1 [13]:

∞∫0

f(y) dq′y = 1 − q′

1 − (q′)1/(d+1)

∞∫0

f(ud+1)ud d(q′)1/(d+1)u,

where f(y) = yk−1/2Gd+1,00,d+1

[ηy; q′| −

b1, . . . , bd+1

], we obtain the desired result. �

3.3. Components

Douak and Maroni [12] proved that {Pn}n�0 is d-symmetric if and only if there exist (d + 1) sequences{P l

n}n�0, 0 � l � d; called the (d + 1) components of {Pn}n�0; such that Pn(d+1)+l(x) = xlP ln(xd+1), and

if {Pn}n�0 is a d-OPS, then {P ln}n�0 are, also, d-OPSs.

Our purpose in this subsection is to determine the (d + 1) components of {Pi,n}n�0, i = 1, 2.

Theorem 3.5. The components {P li,n}n�0, 0 � l � d, of the polynomials {Pi,n}n�0 are d-OPSs of Brenke

type and they have the following generating functions and hypergeometric representations:

exp(at)∏lk=1 vkmk

0Fd

(−m1 + 1, . . . ,ml + 1,ml+1, . . . ,md

; xt∏dk=0 vk

)=

∞∑n=0

P l1,n(x)tn.

P l1,n(x) = an

n!∏l

k=1 vkmk

1Fd

(−n

m1 + 1, . . . ,ml + 1,ml+1, . . . ,md; −x

a∏d

k=0 vk

). (3.12)

eq(at)∏lk=1 uk(ηk − 1)

0Φd

(−η1q, . . . , ηlq, ηl+1, . . . , ηd

; q, ql+1xt∏dk=0 uk

)=

∞∑n=0

P l2,n(x)tn.

P l2,n(x) = an∏l 1Φd

(q−n

η q, . . . , η q, η , . . . , η; q, qn+l+1x∏d

). (3.13)

[n]q! k=1 uk(ηk − 1) 1 l l+1 d a(1 − q) k=0 uk

746 Y. Ben Cheikh, N. Ben Romdhane / J. Math. Anal. Appl. 416 (2014) 735–747

Proof. Using the generating function of the polynomials {P1,n}n�0, we obtain the first result.Replacing the functions exp and 0Fd with their definition expressions, multiplying the sums and using

(n− k)! = (−1)kn!(−n)k and (−n)k = 0, k � n + 1, we obtain the second result.

Similarly, we have the proofs of the two other results replacing exp and 0Fd, respectively, by eq and 0Φd

and using [n− k]q! = (q−1)k

qnk−

(k2) [n]q!

(q−n;q)k and (q−n; q)k = 0, k � n + 1. �

3.4. Remarks

We note that the component polynomials {P l1,n}n�0 were first studied by the first author and Douak [5].

In fact, they obtained that these polynomials are the only d-orthogonal polynomials generated by

exp(t)ψ(xt) =∞∑

n=0Pn(x)tn,

and they were called the Laguerre d-orthogonal type polynomials, {l(αd)n }n�0:

anl(αd)n (x) = n!

l∏k=1

vkPl1,n

(−a

d∏k=0

vkx

),

where the parameters (αd) are chosen as follows: (αd) = (m1, . . . ,ml,ml+1 − 1, . . . ,md − 1).The component polynomials {P l

2,n}n�0 can be expressed by means of the polynomials, {l(αd)n,q }n�0

l(αd)n,q (x) = 1Φd

(q−n

α1, . . . , . . . , αd; q, qnx

). (3.14)

For d = 1, {P l1,n}n�0 are the classical Laguerre polynomials {L(α)

n }n�0 and {P l2,n}n�0 are the little

q-Laguerre polynomials {pn(x, a|q)}n�0 [17]. In fact, we have:

P 02,n

(au0u1η

(1 − q)q

x

)= an(1 − q)n

(q; q)npn

(x, η−1q

∣∣q−1).Concluding remark. In this paper, we determine the d-symmetric d-orthogonal polynomials of Brenke type.We obtain two new families of polynomials that include well known examples in the literature in bothcases d = 1 and d � 1. Some properties for the obtained polynomials are given: The inversion formulas,the moments, the corresponding vectors of weights, providing of integral representations for the moments,and explicit expressions for the components, which provide two sets of d-orthogonal polynomials of Brenketype.

But, the general question that remains not completely solved is:

(P): Find all d-orthogonal PSs of Brenke type.

Acknowledgment

The authors thank the referee for his/her careful reading of the manuscript and for the useful commentsand suggestions.

Y. Ben Cheikh, N. Ben Romdhane / J. Math. Anal. Appl. 416 (2014) 735–747 747

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