# Moments for Generating Functions of Al-Salam-Carlitz Polynomials

Authors: Jian Cao
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We employ the moment representations for Al-Salam-Carlitz polynomials and show how to deduce bilinear, trilinear, and multilinear generating functions for Al-Salam-Carlitz polynomials. Moreover, we obtain two terminating generating functions for Al-Salam-Carlitz polynomials by the method of moments.


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## Body

## 1. Introduction

The Al-Salam-Carlitz polynomials [1, equations (4.1) and (4.2)]
(1.1)∑n=0∞‍Un(a)(x)tn(q;q)n=(t,at;q)∞(xt;q)∞,∑n=0∞‍Vn(a)(x)tn(q;q)n=(xt;q)∞(t,at;q)∞
are important q-orthogonal polynomials whose applications varies in many fields such as the q-harmonic oscillator, theta function, quantum groups, and coding theory [2–4]. The generalized Al-Salam-Carlitz polynomials Un(x,y,a∣q) [5, equation (1.2)] and Vn(x,y,a∣q) are defined by the following
(1.2)∑n=0∞‍Un(x,y,a∣q)tn(q;q)n=(at,yt;q)∞(xt;q)∞,∑n=0∞‍Vn(x,y,a∣q)tn(q;q)n=(xt;q)∞(at,yt;q)∞,
where Un(x,1,a∣q)=Un(a)(x), Vn(x,1,a∣q)=Vn(a)(x) and
(1.3)Un(x,y,a∣q)=(-a)nq(n2)2ϕ1[q-n,yx0;q,qxa],Vn(x,y,a∣q)=an2ϕ0[q-n,xy—;q,yqna].
There are close relationships between Al-Salam-Carlitz polynomials and other polynomials, such as q-Bessel polynomials [6], Stieltjes-Wigert polynomials [7], and Rogers-Szegö polynomials [5]. For more information, please refer to [5–7].Al-Salam and Carlitz [1] defined moments of two discrete distribution dα(a)(x) and dβ(a)(x) by Rogers-Szegö polynomials as follows
(1.4)∫-∞∞‍xndα(a)(x)=hn(a∣q)≜Vn(0,1,a∣q),∫-∞∞‍xndβ(a)(x)=gn(a∣q)≜(-1)nq(n2)Un(0,1,a∣q),
where α(a)(x) is a step function whose jumps occur at the points qk and aqk for k∈ℕ, while the jumps of β(a)(x) occur at the points q-k for k∈ℕ. These jumps are given by the following
(1.5)dα(a)(qk)=qk(a;q)∞(q,q/a;q)k,dα(a)(aqk)=qk(1/a;q)∞(q,aq;q)k,dβ(a)(q-k)=akqk2(aqk+1;q)∞(q;q)k.In view of the fact that the discrete probability measureα(a) of moments in (1.4) on [a,1] may be given by the sum of two terms [8, equation (3.3)],
(1.6)α(a)=∑k=0∞‍[qk(a;q)∞(q,q/a;q)kɛqk+qk(1/a;q)∞(q,aq;q)kɛaqk].
However, β(a) is given by [8, equation (3.15)],
(1.7)β(a)=(aq;q)∞∑k=0∞‍akqk2(aq,q;q)kɛq-k,
where k∈ℕ and ɛy denotes a unit mass supported at y, see also [9, 10].Liu gained the following expression of bivariate Rogers-Szegö polynomials by the technique of partial fraction [11, equation (4.20)],
(1.8)hn(a,b∣q)=an(b/a;q)∞∑k=0∞‍q(n+1)k(q,aq/b;q)k+bn(a/b;q)∞∑k=0∞‍q(n+1)k(q,qb/a;q)n.
So it's natural to define the generalized discrete probability measure α(a,b) and β(a,b) by the following
(1.9)α(a,b)=∑k=0∞‍[qk(a/b;q)∞(q,qb/a;q)kɛbqk+qk(b/a;q)∞(q,aq/b;q)kɛaqk],(1.10)β(a,b)=(aqb;q)∞∑k=0∞‍akb-kqk2(aq/b,q;q)kɛbq-k,
where the bivariate Rogers-Szegö polynomials expressed by the following
(1.11)hn(a,b∣q)=∫-∞+∞‍xndα(a,b)(x)≜Vn(0,a,b∣q),gn(a,b∣q)=∫-∞+∞‍xndβ(a,b)(x)≜Un(0,a,b∣q),
and their generating functions given by [11, equations (2.3) and (2.13)],
(1.12)∑n=0∞‍hn(a,b∣q)tn(q;q)n=1(at,bt;q)∞,∑n=0∞‍gn(a,b∣q)(-1)nq(n2)tn(q;q)n=(at,bt;q)∞.Al-Salam and Ismail [9] built relations between multilinear generating functions for Rogers-Szegö polynomials and 8W7 by the method of moments. Berg and Ismail [8] gained orthogonality relations of generating functions for Rogers-Szegö polynomials by moments. Ismail and Masson [12] obtained several q-beta integrals and Bailey's 6ψ6 from q-Hermite polynomials by moments. Ismail and Stanton [13, 14] derived generating functions for Al-Salam-Chihara and continuous q-Hermite polynomials by moments. For more information, please refer to [8, 9, 12–18].There are many ways to explore generating functions for Al-Salam-Carlitz polynomials. Carlitz [19] used combinatorial method. Srivastava and Jain [20] utilized the technique of transformation theory. Kim [21] obtained combinatorial interpretation by moments. Wang [22] employed the q-integral representation. Chen et al. [5] applied q-exponential operator. For more information, please refer to [2, 5, 19–22].One may ask naturally a question: can Al-Salam-Carlitz polynomials be expressed by moments and some related problems deduced by moments? In this paper, we would like to represent Al-Salam-Carlitz polynomials in terms of moments (see Theorem2.3 below) and show how to deduce generating functions for Al-Salam-Carlitz polynomials by the method of moments.This paper is organized as follows. In Section3, two bilinear generating functions for Al-Salam-Carlitz polynomials are obtained by the method of moments. In Section 4, a trilinear generating function for Al-Salam-Carlitz polynomials is gained and a corollary is achieved. In Section 5, a multilinear generating function for Al-Salam-Carlitz polynomials is supplied naturally. In Section 6, two miscellaneous generating functions for Al-Salam-Carlitz polynomials are given.


## 2. Notations and Moments for Al-Salam-Carlitz Polynomials

In this paper, we follow the notations and terminology in [23] and suppose that 0<q<1. The q-series and its compact factorials are defined, respectively, by the following
(2.1)(a;q)0=1,(a;q)n=∏k=0n-1‍(1-aqk),(a;q)∞=∏k=0∞‍(1-aqk)
and (a1,a2,…,am;q)n=(a1;q)n(a2;q)n…(am;q)n, where m is a positive integer and n is a nonnegative integer or ∞. In the context, convergence of basic hypergeometric series is no issue at all because they are the terminating q-series.Theq-Chu-Vandermonde formula reads that [23, equations (II.6) and (II.7)],
(2.2)2ϕ1[q-n,ac;q,q]=(c/a;q)n(c;q)nan,2ϕ1[q-n,ac;q,cqna]=(c/a;q)n(c;q)n.Theq-difference operators Da and θa defined by [23, 24],
(2.3)Da{f(a)}=f(a)-f(aq)a,θa{f(a)}=f(aq-1)-f(a)aq-1.Before the proof of main results, we need the following lemmas.Lemma 2.1 (q-Leibniz formula [25, 26]).
Forn∈ℕ, one has
(2.4)θan{f(a)g(a)}=∑k=0n‍[nk]θak{f(a)}θan-k{g(aq-k)},(2.5)Dan{f(a)g(a)}=∑k=0n‍[nk]qk(k-n)Dak{f(a)}Dan-k{g(aqk)}.Lemma 2.2 (equations(III.12) and (III.13)).
Forn∈ℕ, one has
(2.6)3ϕ2[q-n,b,cd,e;q,q]=(e/c;q)ncn(e;q)n3ϕ2[q-n,c,dbd,cq1-ne;q,bqe],(2.7)3ϕ2[q-n,b,cd,e;q,deqnbc]=(e/c;q)n(e;q)n3ϕ2[q-n,c,dbd,cq1-ne;q,q],
for d→0 in (2.6) and d→∞ in (2.7), one has
(2.8)3ϕ2[q-n,b,c0,e;q,q]=(e/c;q)ncn(e;q)n2ϕ1[q-n,ccq1-ne;q,bqe],(2.9)3ϕ1[q-n,b,ce;q,eqnbc]=(e/c;q)n(e;q)n2ϕ1[q-n,ccq1-ne;q,qb].Theorem 2.3.
Forn∈ℕ, one has
(2.10)Un(x,y,a∣q)=(-1)nq(n2)(qx/y,qx/a;q)∞∫-∞∞‍μn(qxμay;q)∞dβ(a,y)(μ),(2.11)Vn(x,y,a∣q)=(xy,xa;q)∞∫-∞∞‍μn(xμ/(ay);q)∞dα(a,y)(μ),
where α(y,a)(μ) and β(y,a)(μ) are defined by (1.6) and (1.7).Theorem 2.4.
Forf∈ℕ and s/r=q-f, if max{|ar|,|at|,|bt|,|aw|,|bw|}<1, one has
(2.12)∫-∞∞‍(sμ;q)∞(rμ,tμ,wμ;q)∞dα(a,b)(μ)=(as,abtw;q)∞(ar,at,bt,aw,bw;q)∞3ϕ2[sr,at,awas,abtw;q,br].
Forr/s=q-f, if max{|as|,|abtw/q|}<1, one has
(2.13)∫-∞∞‍(rμ,tμ,wμ;q)∞(sμ;q)∞dβ(a,b)(μ)=(ar,at,bt,aw,bw;q)∞(as,abtw/q;q)∞3ϕ2[rs,q(at),q(aw)qas,q2abtw;q,q].Proof of Theorem2.3.
By (1.11) and (1.12), we have
(2.14)∫-∞∞‍1(tμ;q)∞dα(a,b)(μ)=1(at,bt;q)∞.
Differentiating by Dtn and using q-Leibinz formula (2.5) for Dt on both sides of (2.14), we have
(2.15)∫-∞∞‍μn(tμ;q)∞dα(a,b)(μ)=an(at,bt;q)∞2ϕ0[q-n,at—;q,bqna].
Comparing (1.2), (1.3), and (2.15), we have (2.11). Similarly, by q-Leibniz formula (2.4) for θt, we obtain
(2.16)∫-∞∞‍μn(tμ;q)∞dβ(a,b)(μ)=an(at,bt;q)∞2ϕ1[q-n,q(at)0;q,bt],
so we achieve (2.10) by (1.3). The proof is complete.Proof of Theorem2.4.
Multiplyingwn/(q;q)n and summing n over 0≤n≤∞ on both sides of (2.15) yields
(2.17)∫-∞∞‍1(tμ,wμ;q)∞dα(a,b)(μ)=(abtw;q)∞(at,bt,aw,bw;q)∞.
Differentiating by Dtn and using (2.5) on both sides of (2.17), we have
(2.18)Dtn{(abtw;q)∞(at,bt;q)∞}=∑k=0n‍qk(k-n)[nk]Dtk{(abtw;q)∞(bt;q)∞}Dtn-k{1(atqk;q)∞}=∑k=0n‍qk(k-n)[nk]bk(aw;q)k(abtwqk;q)∞(bt;q)∞(aqk)n-k(atqk;q)∞,
that is
(2.19)∫-∞∞‍μn(tμ,wμ;q)∞dα(a,b)(μ)=an(abtw;q)∞(at,bt,aw,bw;q)∞3ϕ1[q-n,aw,atabtw;q,bqna].
Multiplying (s/r;q)nrn/(q;q)n and summing on both sides of (2.19), we obtain (2.12) after simplification. Similarly, we have
(2.20)∫-∞∞‍(tμ,wμ;q)∞dβ(a,b)(μ)=(at,bt,aw,bw;q)∞(abtw/q;q)∞,
by q-Leibniz formula for θt, we gain
(2.21)θtn{(at,bt;q)∞(abtw/q;q)∞}=∑k=0n‍[nk]q-(k+12)(abw)k(q/(aw);q)k(bt;q)∞(abtwq-1-k;q)∞(-aq-k)n-k(atq-k;q)∞,
so we have
(2.22)∫-∞∞‍μn(tμ,wμ;q)∞dβ(a,b)(x)=an(aw,bw,at,bt;q)∞(abtw/q;q)∞3ϕ2[q-n,q(aw),q(at)q2(abtw),0;q,q],
which is (2.13) after multiplying (r/s;q)nsn/(q;q)n and summing on both sides of (2.22). The proof is complete.


## 3. Bilinear Generating Function for Al-Salam-Carlitz Polynomials

Chen et al. [5] gave the following bilinear generating functions for Al-Salam-Carlitz polynomials.Theorem 3.1 (see [5, Theorem 5.3]).
Assume thatmax{|xq/a|,|uq/b|}<1. One has
(3.1)∑k=0∞‍(-1)m+n+kq-(n+k2)-(m2)-mkUn+k(x,y,a∣q)Um+k(u,v,b∣q)zk(q;q)k=(yq/a,vq/b,abz;q)∞anbm(xq/a,uq/b;q)∞∑k=0∞‍(y/x,q/(abz);q)k(q,yq/a;q)k(xbzqn)k×2ϕ1[vu,q1+k(abz)vqb;q,auzqm+k],
where y/x=q-f and v/u=q-g for nonnegative integers f,g and |auzq-(m+k)|<1.In this section, we gain (3.1) and the following dual ones by the method of moments.Theorem 3.2.
Assume thatmax{|yq/a|,|vq/b|}<1. One has
(3.2)∑k=0∞‍Vn+k(x,y,a∣q)Vm+k(u,v,b∣q)(zq)k(q;q)k=(x/a,u/b;q)∞anbm(y/a,abz,v/b;q)∞∑k=0∞‍(x/y,abz;q)kq(1+n)k(q,aq/y;q)k×2ϕ1[uv,abzqkbqv;q,q1+m],
where x/y=q-f and u/v=q-g for nonnegative integers f,g.Corollary 3.3 (see [5, Theorem 4.5]).
Forf∈ℕ, one has
(3.3)∑k=0∞‍(-1)kq-(k2)Uk(x,y,a∣q)Uk(u,v,b∣q)(zq)k(q;q)k=(abzq,avzq,yq/a;q)∞(auzq,xq/a;q)∞3ϕ2[yx,1(abz),1(avz)1(auz),yqa;q,xbvzqu],
provided that y/x=q-f or v/u=q-f and max{|auzq-f|,|xq/a|,|xbvz/u|}<1.Corollary 3.4.
Forf∈ℕ, one has
(3.4)∑k=0∞‍Vk(x,y,a∣q)Vk(u,v,b∣q)zk(q;q)k=(auz,x/a;q)∞(abz,avz,y/a;q)∞3ϕ2[abz,xy,avzauz,aqy;q,q],
provided that x/y=q-f or u/v=q-f and max{|abz|,|avz|,|y/a|}<1.Remark 3.5.
Form=n=0, formula (3.1) and (3.2) reduce to (3.3) and (3.4), respectively, by q-Gauss sum [23, equation (II.8)],
(3.5)2ϕ1[a,bc;q,cab]=(c/a,c/b;q)∞(c,c/(ab);q)∞
and q-Chu-Vandermonde formula (2.2).Proof of Theorems3.1 and 3.2.
The left-hand side of (3.1) is equal to
(3.6)∑k=0∞‍(-1)m+n+kq-(n+k2)-(m2)-mk(-1)n+kq(n+k2)(qx/y,qx/a;q)∞∫-∞∞‍μn+k(qxμ(ay);q)∞dβ(a,y)(μ)×(-1)m+kq(m+k2)(qu/v,qu/b;q)∞∫-∞∞‍wm+k(quw(bv);q)∞dβ(b,v)(w)zk(q;q)k=1(qx/y,qx/a,qu/v,qu/b;q)∞∬-∞∞μnwm(qxμ(ay),quw(bv),μwz;q)∞dβ(a,y)(μ)dβ(b,v)(w)=1(qu/v,qu/b;q)∞∫-∞∞‍{1(qx/y,qx/a;q)∞∫-∞∞‍μn(qxμ(ay),μwz;q)∞dβ(a,y)(μ)}×wm(quw(bv);q)∞dβ(b,v)(w)=1(qu/v,qu/b;q)∞∫-∞∞‍{an(awz,ywz;q)∞(xwz;q)∞3ϕ2[q-n,q(awz),yxq(xwz),0;q,q]}by(2.8)×wm(quw(bv);q)∞dβ(b,v)(w)=1(qu/v,qu/b;q)∞∫-∞∞‍{[yx,qawzyqa;q,xwzqn][yx,q(awz)yqa;q,xwzqn]an(awz,ywz;q)∞(xwz;q)∞(qy/a,xwz;q)∞(qx/a,ywz;q)∞×2ϕ1[yx,q(awz)yqa;q,xwzqn]}×wm(quw(bv);q)∞dβ(b,v)(w)=an(qy/a;q)∞(qu/v,qu/b,qx/a;q)∞∑k=0∞‍(y/x;q)k(-1)kq(k+12)(q,yq/a;q)k(xaqn)k×∫-∞∞‍wm(quw(bv),awzq-k;q)∞dβ(b,v)(w)by(2.22)=an(qy/a;q)∞(qu/v,qu/b,qx/a;q)∞∑k=0∞‍(y/x;q)k(-1)kq(k+12)(q,yq/a;q)k(xaqn)k×bm(qu/b,qu/v,abzq-k,avzq-k;q)∞(auzq-k;q)∞×3ϕ2[q-m,q1+k(abz),vuq1+k(auz),0;q,q]by(2.8)=an(qy/a;q)∞(qu/v,qu/b,qx/a;q)∞∑k=0∞‍(y/x;q)k(-1)kq(k+12)(q,yq/a;q)k(xaqn)k×bm(qu/b,qu/v,abzq-k,avzq-k;q)∞(auzq-k;q)∞(qv/b,auzq-k;q)∞(uq/b,avzq-k;q)∞2ϕ1[vu,q1+k(abz)qvb;q,auzqm+k],
which is the right hand side of (3.1) after simplification. Similarly, the right hand side of (3.2) is equivalent to
(3.7)∑k=0∞‍(xy,xa;q)∞∫-∞∞‍μn+k(xμ/(ay);q)∞dα(a,y)(μ)(uv,ub;q)∞×∫-∞∞‍wm+k(uw/(bv);q)∞dα(b,v)(w)(zq)k(q;q)k=(uv,ub;q)∞∫-∞∞‍{(xy,xa;q)∞∫-∞∞‍μn(xμ/(ay),μwz;q)∞dα(a,y)(μ)}×wm(uw/(bv);q)∞dα(b,v)(w)=(uv,ub;q)∞∫-∞∞‍an(xwz;q)∞(awz,ywz;q)∞3ϕ1[q-n,awz,xyxwz;q,yqna]×wm(uw/(bv);q)∞dα(b,v)(w)by(2.9)=(uv,ub;q)∞∫-∞∞‍an(xwz;q)∞(awz,ywz;q)∞(x/a,ywz;q)∞(xwz,y/a;q)∞2ϕ1[xy,awzaqy;q,q1+n]×wm(uw/(bv);q)∞dα(b,v)(w)=an(u/v,u/b,x/a;q)∞(y/a;q)∞∑k=0∞‍(x/y;q)kq(1+n)k(q,aq/y;q)k×∫-∞∞‍wm(uw/(bv),awzqk;q)∞dα(b,v)(w)by(2.19)=an(u/v,u/b,x/a;q)∞(y/a;q)∞∑k=0∞‍(x/y;q)kq(1+n)k(q,aq/y;q)kbm(auzqk;q)∞(u/v,u/b,abzqk,avzqk;q)∞×3ϕ1[q-m,abzqk,uvauzqk;q,vqmb],
which is the right hand side of (3.2) after using (2.9). The proof is complete.


## 4. Trilinear Generating Function for Al-Salam-Carlitz Polynomials

Verma and Jain deduced the following trilinear generating functions for Rogers-Szegö polynomials.Proposition 4.1 (see [27, equation (2.14)]).
Fors∈ℕ, one has
(4.1)∑n=0∞‍hn+s(x∣q)hn(y∣q)hn(z∣q)wn(q;q)n=(yzw2;q)∞(x,w,yw,zw,yzw;q)∞6ϕ5[w,yw,zw,yzw,0,0qx,wyz,-wyz,wyzq,-wyzq;q,q1+s]+xs(yzx2w2;q)∞(1/x,xw,xyw,xzw,xyzw;q)∞×6ϕ5[xw,xyw,xzw,xyzw,0,0xq,xwyz,-xwyz,xwyzq,-xwyzq;q,q1+s].In this section, we obtain the following trilinear generating functions for Al-Salam-Carlitz polynomials.Theorem 4.2.
Fork∈ℕ, one has
(4.2)∑n=0∞‍Vn(x,y,a∣q)Vn(u,v,b∣q)Vn+k(r,s,c∣q)wn(q;q)n=(r/s,auws,bxws;q)∞sk(c/s,abws,avws,byws;q)∞∑j=0∞‍(r/c,abws,avws,byws;q)jq(1+k)j(q,qs/c,auws,bxws;q)j×3ϕ2[xy,uv,abwsqjauwsqj,bxwsqj;q,yvwsqj]+(r/c,auwc,bxwc;q)∞ck(s/c,abwc,avwc,bywc;q)∞×∑j=0∞‍(r/s,abwc,avwc,bywc;q)∞q(1+k)j(q,qc/s,auwc,bxwc;q)j3ϕ2[xy,uv,abwcqjauwcqj,bxwcqj;q,yvwcqj],
provided that the right hand side of (4.2) is convergent.Remark 4.3.
Ismail and Stanton [14] developed a method for deriving integral representations of certain orthogonal polynomials as moments and obtained [14, equation (4.13)] trilinear generating functions for q-Hermite polynomials Hn(cosθ∣q), which is equivalent to Proposition 4.1. For more information, please refer to [14].Proof of Theorem4.2.
The left hand side of (4.2) is equal to
(4.3)(rs,rc;q)∞∑n=0∞‍∫-∞∞‍μn+k(rμ/(cs);q)∞dα(c,s)(μ)Vn(x,y,a∣q)Vn(u,v,b∣q)wn(q;q)n=(rs,rc;q)∞∫-∞∞‍μk(rμ/(cs);q)∞∑n=0∞‍Vn(x,y,a∣q)Vn(u,v,b∣q)(wμ)n(q;q)ndα(c,s)(μ)by(3.4)=(rs,rc;q)∞∫-∞∞‍μk(auwμ,x/a;q)∞(rμ/(cs),abwμ,avwμ,y/a;q)∞×3ϕ2[abwμ,xy,avwμauwμ,aqy;q,q]dα(c,s)(μ)=(rs,rc;q)∞∫-∞∞‍μk(auwμ,x/a;q)∞(rμ/(cs),abwμ,avwμ,y/a;q)∞×(bxwμ,y/a;q)∞(bywμ,x/a;q)∞3ϕ2[xy,uv,abwμauwμ,bxwμ;q,yvwμ]dα(c,s)(μ)by(1.9)=(rs,rc;q)∞∑j=0∞‍qj(c/s;q)∞(q,qs/c;q)j(sqj)k(auwsqj,bxwsqj;q)∞(rqj/c,abwsqj,avwsqj,bywsqj;q)∞×3ϕ2[xy,uv,abwsqjauwsqj,bxwsqj;q,yvwsqj]+(rs,rc;q)∞∑j=0∞‍qj(s/c;q)∞(q,qc/s;q)j×(cqj)k(auwcqj,bxwcqj;q)∞(rqj/s,abwcqj,avwcqj,bywcqj;q)∞3ϕ2[xy,uv,abwcqjauwcqj,bxwcqj;q,yvwcqj],
which is the right hand of (4.2) after simplification. The proof is complete.Proof of Proposition4.1.
Noting the fact thatVn(0,y,1∣q)=hn(y∣q) and
(4.4)(vw2y;q)2j=(wvy,-wvy,wvyq,-wvyq;q)j,(s2vw2y;q)2j=(swvy,-swvy,swvyq,-swvyq;q)j.
If letting x=u=r=0 and a=b=c=1 in (4.2), the right hand side of (4.2) is equal to
(4.5)sk(s2vw2y;q)∞(1/s,ws,vws,yws,yvws;q)∞∑j=0∞‍(ws,vws,yws,yvws;q)jq(1+k)j(q,qs;q)j(s2vw2y;q)2j+(vw2y;q)∞(s,w,vw,yw,yvw;q)∞∑j=0∞‍(w,vw,yw,yvw;q)j(q,q/s;q)j(vw2y;q)2j,
which reduces to the right hand side of (4.1) after setting (s,v)=(x,z). The proof is complete.


## 5. Multilinear Generating Functions for Al-Salam-Carlitz Polynomials

Ismail and Stanton gained the following multilinear generating functions for Rogers-Szegö polynomials.Proposition 5.1 (see [28, equation (5.14)]).
One has(5.1)∑m1,…,mk=0∞‍hm1+⋯+mk(a∣q)hm1(t1t2∣q)⋯hmk(t2k-1t2k∣q)∏j=1k‍t2jmj(q;q)mj=2kϕ2k-1[t1,…,t2kq/a,0,…,0;q,q]{(a;q)∞∏j=12k(tj;q)∞}+2kϕ2k-1[at1,…,at2kaq,0,…,0;q,q]{(1/a;q)∞∏j=12k(atj;q)∞}.In this section, we obtain the following multilinear generating functions for Al-Salam-Carlitz polynomials.Theorem 5.2.
Fors∈ℕ, one has
(5.2)∑m1,…,nk=0∞‍Vm1+⋯+mk+s(y,x,a∣q)Vm1(y1,x1,a1∣q)⋯Vmk(yk,xk,ak∣q)t1m1⋯tkmk(q;q)m1⋯(q;q)mk=as(y/a,y1t1a,…,yktka;q)∞(x/a,a1t1a,x1t1a,…,aktka,xktka;q)∞×2k+1ϕ2k[yx,a1t1a,x1t1a,…,aktka,xktkaaqx,y1t1a,…,yktka;q,q1+s]+xs(y/x,y1t1x,…,yktkx;q)∞(a/x,a1t1x,x1t1x,…,aktkx,xktkx;q)∞×2k+1ϕ2k[ya,a1t1x,x1t1x,…,aktkx,xktkxxqa,y1t1x,…,yktkx;q,q1+s],
provided that the right hand side of (5.2) is convergent.Corollary 5.3 (see [20, equation (1.12)]).
Fors∈ℕ, one has
(5.3)∑n1,…,nk=0∞‍hn1+⋯+nk+s(x∣q)hn1(x1∣q)…hnk(xk∣q)t1n1(q;q)n1⋯tknk(q;q)nk=1(x,t1,x1t1,…,tk,xktk;q)∞2kϕ2k-1[t1,x1t1,…,tk,xktkqx,0,…,0;q,q1+s]+xs(1/x,xt1,xx1t1,…,xtk,xxktk;q)∞2kϕ2k-1[xt1,xx1t1,…,xtk,xxktkqx,0,…,0;q,q1+s].Remark 5.4.
Lettingy=y1=⋯=yk=s=0, a=a1=⋯=ak=1 and taking (x,xi,ti)=(a,ti/ti+1,t2i) in Theorem 5.2, formula (5.2) reduces to (5.1). Setting y=y1=⋯=yk=0 and a=a1=⋯=ak=1 in Theorem 5.2, formula (5.2) reduces to (5.3).Proof.
The left hand side of (5.2) is equal to
(5.4)(yx,ya;q)∞∫-∞∞‍∑m1,…,nk=0∞‍Vm1(y1,x1,a1∣q)⋯Vmk(yk,xk,ak∣q)×(t1μ)m1⋯(tkμ)mk(q;q)m1⋯(q;q)mkμsdα(a,x)(μ)(yμ/(ax);q)∞=(yx,ya;q)∞∫-∞∞‍μs(y1t1μ,…,yktkμ;q)∞(a1t1μ,x1t1μ,…,aktkμ,xktkμ,yμ/(ax);q)∞dα(a,x)(μ)=(yx,ya;q)∞[∑r=0∞‍qr(x/a;q)∞(q,qa/x;q)r×(aqr)s(y1t1aqr,…,yktkaqr;q)∞(a1t1aqr,x1t1aqr,…,aktkaqr,xktkaqr,yqr/x;q)∞+∑r=0∞‍qr(a/x;q)∞(q,qx/a;q)r×(xqr)s(y1t1xqr,…,yktkxqr;q)∞(a1t1xqr,x1t1xqr,…,aktkxqr,xktkxqr,yqr/a;q)∞]=as(y/a,y1t1a,…,yktka;q)∞(x/a,a1t1a,x1t1a,…,aktka,xktka;q)∞×2k+1ϕ2k[yx,a1t1a,x1t1a,…,aktka,xktkaaqx,y1t1a,…,yktka;q,q1+s]+xs(y/x,y1t1x,…,yktkx;q)∞(a/x,a1t1x,x1t1x,…,aktkx,xktkx;q)∞,×2k+1ϕ2k[ya,a1t1x,x1t1x,…,aktkx,xktkxxqa,y1t1x,…,yktkx;q,q1+s],
which is the right hand side of (5.2) after simplification. The proof is complete.


## 6. Some Miscellaneous Generating Functions for Al-Salam-Carlitz Polynomials

In this section, we deduce the following terminating generating functions for Al-Salam-Carlitz polynomials.Theorem 6.1.
Forn∈ℕ, one has
(6.1)∑k=0n‍Uk(cqk-1,d,b∣q)(-1)kqnk-(k2)(q-n,a;q)k(q,c/d,c/b;q)k(cabd)k=(c/(ad);q)n(c/d,c/b;q)n2ϕ1[q-n,dq1-ncadq1-nc;q,cqnb].Theorem 6.2.
Forn∈ℕ, one has
(6.2)∑k=0n‍Vn-k(aqk,d,b∣q)(q-n,a/d,a/b;q)k(q,c;q)k(bcdqna)k=(cd/a;q)nbn(c;q)n2ϕ1[q-n,adaq1-n(cd);q,aqbc].Corollary 6.3.
Forn∈ℕ, one has
(6.3)∑k=0n‍gk(d∣q)(q-n;q)k(q;q)k(cqnd)k=(cd;q)n1ϕ1[q-ndq1-nc;q,dq],∑k=0n‍hn-k(d∣q)(q-n;q)k(q;q)k(cdqn)k=(cd;q)n2ϕ1[q-n,0q1-n(cd);q,qc].Remark 6.4.
Lettingb→∞ in Theorems 6.1 and 6.2, noting the fact that
(6.4)lim⁡b→∞(-1)kq-(k2)b-kUk(cqk-1,d,b∣q)=lim⁡b→∞bk-nVn-k(aqk,d,b∣q)=1.
Formula (6.1) and (6.2) reduce to q-Chu-Vandermonde formula (2.2), respectively. Replacing c by ac and letting (a,b)=(0,1) in Theorems 6.1 and 6.2, formula (6.1) and (6.2) reduce to (6.3), respectively.Proof of Theorems6.1 and 6.2.
Replacingc by cμ/(bd) on both sides of the second formula in (2.2), we have
(6.5)∑k=0n‍(q-n,a;q)k(q;q)k(cqnabd)k⋅μk(cμqk(bd);q)∞=(cμ/(abd),cμqn/(bd);q)∞(cμqn/(bd);q)∞.
Applying moments ∫-∞∞‍dβ(b,d)(μ) in (6.5), using (2.10) and (2.13), we gain
(6.6)∑k=0n‍(q-n,a;q)k(q;q)k(cqnabd)k(-1)kq-(k2)(cqkd,cqkb;q)∞Uk(cqk-1,d,b∣q)=(c/(ad),cqn/d,cqn/b;q)∞(cqn/(ad);q)∞2ϕ1[q-n,dq1-ncadq1-nc;q,cqnb],
which is (6.1) after simplification. Similarly, setting a by aw/(bd) on both sides of the second formula in (2.2), we obtain
(6.7)∑k=0n‍(q-n;q)k(q,c;q)k(bcdqna)k⋅wn-k(awqk/(bd);q)∞=(-1)nq(n2)(bcd/a)n(c;q)n⋅(awq1-n/(bcd);q)∞(awq/(bcd),aw/(bd);q)∞.
Utilizing moments ∫-∞∞‍dα(b,d)(w) in (6.7), by (2.11) and (2.12), we have
(6.8)∑k=0n‍(q-n;q)k(q,c;q)k(bcdqna)k1(aqk/d,aqk/b;q)∞Vn-k(aqk,d,b∣q)=(-1)nq(n2)(bcd/a)n(c;q)n(aq1-n/(cd);q)∞(aq/(cd),a/d,a/b;q)∞2ϕ1[q-n,adaq1-n(cd);q,aqbc],
which becomes (6.2) after simplification. The proof is complete.



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*Source: 548168-2012-08-13.xml*