# Some Integrals Involvingq-Laguerre Polynomials and Applications

Authors: Jian Cao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The integrals involving multivariateq-Laguerre polynomials and  then auxiliary ones are studied. In addition, the representations of q-Hermite polynomials by q-Laguerre polynomials and their related integrals are given. At last, some generalized integrals associated with generalized q-Hermite polynomials are deduced.


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## Body

## 1. Introduction

Theq-Laguerre polynomials are important q-orthogonal polynomials whose applications and generalizations arise in many applications such as quantum group (oscillator algebra, etc.), q-harmonic oscillator, and coding theory. For example, covariant oscillator algebra can be expressed by q-Laguerre polynomials [1]. The q-deformed radial Schrödinger  is analyzed by q-Laguerre polynomials [2]. The q-Laguerre polynomials are the eigenvectors of an suq(1∣q)-representation by [3]. For more information, please refer to [1–5].Theq-Laguerre polynomials are defined by [6, equation (1.0.1)]
(1)ℒn(α)(x;q)=(qα+1;q)n(q;q)n∑k=0n‍(q-n;q)kq(k2)(xqn+α+1)k(qα+1;q)k(q;q)k,
which belong to the Askey scheme of basic hypergeometric orthogonal polynomials and according to Koekoek and Swarttouw [7, equation (3.21.1)]. The case of x in (1) replaced by (1-q)x is studied by Moak [8, equation (2.3)].In this paper, we first define the auxiliaryq-Laguerre polynomials as follows:
(2)ℳn(α)(x;q)=q-nα(qα+1;q)n(q;q)n∑k=0n‍(q-n;q)kxk(qα+1;q)k(q;q)k.It is easy to see the validity of the following:(3)ℒn(α)((1-q)x;q)=ℳn(α)((1-q-1)x;q-1),(4)limq→1-ℒn(α)((1-q)x;q)=limq→1+ℳn(α)((1-q)x;q)=Ln(α)(x),
where the classical Laguerre polynomials are defined by [9, page 201]
(5)Ln(α)(x)=∑k=0n‍(-1)k(1+α)nxkk!(n-k)!(1+α)k.
For more information about classical Laguerre polynomials, please refer to [9–15] and the references therein.The well-known orthogonality ofq-Laguerre polynomials reads the following.Proposition 1 (see [6, equation (2.0.7)] and [8, equation (2.4)]).
Forα>-1 and for m,n∈ℕ, one has
(6)∫0∞‍ℒm(α)(x;q)ℒn(α)(x;q)xα(-x;q)∞dx=-q-n(qα+1;q)n(q-α;q)∞(q;q)n(q;q)∞πδm,nsinπα.Hahn [16] discovered the previous q-extensions of the Laguerre polynomials, although he said little about them. Moak [8] found that the q-Laguerre polynomials are orthogonal with respect to the discrete measures (Dirac measure). Koekoek and Meijer [17–19] studied systematically the inner product of q-Laguerre polynomials. Ismail and Rahman [20] studied the indeterminate Hamburger moment problems related to q-Laguerre polynomials. For more information, please refer to [6–8, 16–21] and the references therein.In this paper, we first generalize Proposition1 and the auxiliary ones as follows.Theorem 2.
Forℜ{μ}>-1 and m,n∈ℕ, one has
(7)∫0∞‍ℒm(α)(xy;q)ℒn(β)(xz;q)xμ(-x;q)∞dx=(1-q)1+μπcsc(-μπ)Γq(-μ)×∑k=0min{m,n}‍[m+αm-k][n+βn-k][k+μk]q(α+β-2μ-1)kykzk×ϕ21[qk-m,qμ+1+kqα+k+1;yqα-μ+m-k]×ϕ21[qk-n,qμ+1+kqβ+k+1;zqβ-μ+n-k].Theorem 3.
Forℜ{μ}>-1 and m,n∈ℕ, one has
(8)∫0∞‍ℳm(α)(xy;q)ℳn(β)(xz;q)xμ(x;q)∞dx=q(μ+22)-αm-βn(1-q)1+μπcsc(-μπ)Γq(-μ)×∑k=0min{m,n}‍[m+αm-k][n+βn-k][k+μk]×q(2k+μ-m-n+1)kykzk×ϕ21[qk-m,qμ+1+kqα+k+1;yq]ϕ21[qk-n,qμ+1+kqβ+k+1;zq].Corollary 4 (see [15, equation (14)]).
Forℜ{γ}>-1, ℜ{σ}>0, and m,n∈ℕ, one has
(9)∫0∞‍xγe-σxLm(α)(λx)Ln(β)(μx)dx=Γ(γ+1)σγ+1∑k=0min{m,n}‍(m+αm-k)×(n+βn-k)(k+γk)(λμσ2)k×F21[-m+k,γ+k+1α+k+1;λσ]×F21[-n+k,γ+k+1β+k+1;μσ].Remark 5.
Theorems2 and 3 reduce to Proposition 1 and formula (41), respectively, if letting y=z=1 and α=β=μ, and become Corollary 4 by setting q→1 and taking (μ,x,y,z)=(γ,σx,λ/σ,μ/σ).The discreteq-Hermite polynomials hn(x;q) and h~n(x;q) are defined by [7, pages 90-91]
(10)hn(x;q)=xn2ϕ0[q-n,q-n+1-;q2,q2n-1x2]≜q(n2)𝒢n(x1-q2;q2),h~n(x;q)=xn2ϕ1[q-n,q-n+10;q2,q2x2]≜q-(n2)ℋn(x1-q2;q2),
which are equivalent to Al-Salam-Carlitz polynomials with a=-1 (please refer to [22, page 53] also), and the relation between them is hn(ix;q-1)=inh~n(x;q). For more information about the Al-Salam-Carlitz polynomials and the discrete q-Hermite polynomials, please refer to [7, 22–30] and the references therein.In this paper, we also define newq-Hermite polynomials ℋn(x;q) and 𝒢n(x;q), whose names come from the facts
(11)limq→1(1-q)-n/2ℋn((1-q)x;q)=limq→1(1-q)-n/2𝒢n((1-q)x;q)=2-nHn(x),(12)limx→(1-q)x,t→(1-q)t,q→1∑n=0∞‍tn(q1/2;q1/2)nℋn(x;q)=exp(2xt-t2)=∑n=0∞‍tnn!Hn(x)=limx→(1-q)x,t→(1-q)t,q→1∑n=0∞‍tnqn(n-1)/4(q1/2;q1/2)n𝒢n(x;q),
then we deduce the representations of ℋn(x;q) and 𝒢n(x;q) by q-Laguerre polynomials; see Theorems 15 and 16.As an application, using the orthogonality ofq-Laguerre polynomials (6), and (41), and combining the expressions of q-Hermite polynomials (52) and (54), we can obtain the following results immediately.Theorem 6.
Forα>-1 and j≤n∈ℕ, one has
(13)∫0∞‍ℋn(x;q)ℒj(α)(x;q)xα(-x;q)∞dx=-(qα+1,q;q)n(qα+1;q)j(q-α;q)∞(1-q)-n/2(q;q)j(q;q)∞×πC(n,j)sinπα,
where C(n,k) is defined by (52).Theorem 7.
Forα>-1 and j≤n∈ℕ, one has
(14)∫0∞‍𝒢n(x;q)ℳj(α)(x;q)xα(x;q)∞dx=-q-(1/2)(n2)+(α+22)+j-αj×(qα+1,q;q)n(qα+1;q)j(q-α;q)∞(1-q)-n/2(q;q)j(q;q)∞×πD(n,j)sinπα,
where D(n,k) is defined by (54).The generalized Hermite polynomials were introduced by Szegö [31], (see also [23, equation (1.1)]) as follows:
(15)H2n(μ)(x)=(-1)n22nn!Ln(μ-1/2)(x2),H2n+1(μ)(x)=(-1)n22n+1n!xLn(μ+1/2)(x2).The authors [23, equation (2.7)] defined the following generalized q-Hermite polynomials:
(16)ℋ2n(μ)(x;q)=(-1)n(q;q)nℒn(μ-1/2)(x2;q),ℋ2n+1(μ)(x;q)=(-1)n(q;q)nxℒn(μ+1/2)(x2;q)
and deduced their orthogonal relations; see Proposition 19 below.In this paper, we continue to define the auxiliary polynomials according to (16) as follows:
(17)𝒢2n(μ)(x;q)=q-(n+12)(q;q)nℳn(μ-1/2)(x2;q),𝒢2n+1(μ)(x;q)=q-(n+12)(q;q)nxℳn(μ+1/2)(x2;q).With the aid of (15)–(17) and (4), one readily verifies that
(18)limq→1(1-q)-n/2ℋn(μ)(1-qx;q)=limq→1(1-q)-n/2𝒢n(μ)(1-qx;q)=2-nHn(μ)(x).As another application of this paper, we gain the generalq-Laguerre polynomials of several variables by Theorems 2 and 3, and we also deduce the orthogonal polynomials of 𝒢n(μ)(x;q). For more details of the results, see Theorems 20 and 21 and Corollary 23.The structure of this paper is organized as follows. In Section2, we show how to prove the integrals involving q-Laguerre polynomials of several variables. In Section 3, we represent discrete q-Hermite polynomials by q-Laguerre polynomials and their related integral results. In Section 4, we study the general integrals of q-Hermite polynomials involving several variables.


## 2. Notations and Proof of Theorems2 and 3

Throughout this paper, we follow the notations and terminology in [32] and assume that 0<q<1, ℕ={0,1,2,…}, and ℝ is rational number. The q-series and its compact factorials are defined [32, page 6], respectively, by
(19)(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 ∞.The basic hypergeometric seriesϕrs is given by
(20)ϕrs[a1,…,arb1,…,bs;q,z]=∑n=0∞(a1,a2,…,ar;q)n(q,b1,…,bs;q)nzn[(-1)nq(n2)]s+1-r.
For convergence of the infinite series in (20), |q|<1 and |z|<∞ when r≤s, or |q|<1 and |z|<1 when r=s+1, provided that no zeros appear in the denominator. Letting (ai,bi)=(qai,qbi) and setting q→1, (20) reduces to the classical Gauss’ hypergeometric series
(21)Frs[a1,a2,…,arb1,…,bs;z]=∑n=0∞‍(a1)n(a2)n⋯(ar)nn!(b1)n⋯(bs)nzn,
where Pochhammer symbol (z)n is defined by (z)n=z(z+1)⋯(z+n-1)=Γ(z+n)/Γ(z).Theq-analogue of the gamma function is defined by (see [32, equation (1.10.1)]) as follows:
(22)Γq(x)=(q;q)∞(qx;q)∞(1-q)1-x,0<q<1.Theq-Chu-Vandermonde formula [32, equations (II.6) and (II.7)] reads that
(23)ϕ21[q-n,bc;q,q]=(c/b;q)n(c;q)nbn,ϕ21[q-n,bc;q,cqnb]=(c/b;q)n(c;q)n.Theϕ32 transformations [32, equations (III.12) and (III.13)] stats that
(24)ϕ32[q-n,a,cb,d;q,q]=an(d/a;q)n(d;q)n×ϕ32[q-n,a,bcb,aq1-nd;q,cqd],ϕ32[q-n,b,cd,e;q,deqnbc]=(e/c;q)n(e;q)n×ϕ32[q-n,c,dbd,cq1-ne;q,cqd].Theq-analogue of the Pfaff-Kummer transformation [32, equation (III.4)] is as follows:
(25)ϕ21[a,bc;q,z]=(az;q)∞(z;q)∞ϕ22[a,cbc,az;q,bz].The Ramanujan beta integral is stated as follows [33, equation (2.8)]:
(26)∫0∞tx-1(-at;q)∞(-t;q)∞dt=(a,q1-x;q)∞(q,aq-x;q)∞πsinπx,(0<a<qx,x>0).Lemma 8 (see [33, equation (4.2)]).
One has(27)∫0∞‍xα(-x;q)∞dx=Γ(-α)Γ(α+1)(1-q)1+αΓq(-α)=-(q-α;q)∞(q;q)∞πsinπα,(28)∫0∞‍xα(x;q)∞dx=-q(α+22)(q-α;q)∞(q;q)∞πsinπα.Proof.
Taking(a,t)=(-1/a,at) in (26), then letting a→0, we obtain (28) immediately. The proof is complete.Lemma 9.
Forα>-1 and n∈ℕ, one has
(29)ℒn(α)(xy;q)=∑k=0n‍(qα+1;q)n(y;q)n-kykℒk(α)(x;q)(q;q)n-k(qα+1;q)k,(30)ℳn(α)(xy;q)=∑k=0n‍(-1)n-k(qα+1;q)n(1/y;q)n-kynℳk(α)(x;q)(q;q)n-k(qα+1;q)k×q(k2)+αk-(n2)-αn.Proof.
Lettingγ=0 in [6, Proposition 4.1],
(31)∑n=0∞‍(γt;q)nqαn+n2(γt,qα+1,q;q)n(-xt)n=(t;q)∞(γt;q)∞∑n=0∞‍(γ;q)ntn(qα+1;q)nℒn(α)(x;q),
then replacing x by xy, we have
(32)∑n=0∞‍ℒn(α)(xy;q)(qα+1;q)ntn=(yt;q)∞(t;q)∞1(yt;q)∞∑n=0∞‍qαn+n2(qα+1,q;q)n(-xyt)n=∑k=0∞‍(y;q)k(q;q)ktk∑n=k∞‍Ln-k(α)(x;q)(yt)n-k(qα+1;q)n-k.
Comparing the coefficients of tn on both sides of (32) yields (29). Similar to (32), by the definition (1), we have
(33)∑n=0∞‍(γ;q)n(tqα)n(qα+1;q)nℳn(α)(x;q)=(γt;q)∞(t;q)∞∑n=0∞‍(γ;q)nxn(q/t,qα+1,q;q)n.
By taking (γ,t,x)=(1/γ,γt,xy) and letting γ→0 in (33), we obtain (30). The proof of Lemma 9 is complete.Lemma 10.
Formin{α,β}>-1 and m,n∈ℕ, one has
(34)∫0∞‍ℒm(α)(x;q)ℒn(β)(x;q)xβ(-x;q)∞dx=(-1)m+nq(α-β)m+(m2)+(n2)-mn×(1-q)1+β[β-αm-n][n+βn]πcsc(-βπ)Γq(-β),(35)∫0∞‍ℳj(α)(x;q)ℳk(β)(x;q)xβ(x;q)∞dx=(-1)k+jq(j+12)+(k+12)-(α+j)k+(β+22)×(1-q)1+β[β-αj-k][k+βk]πcsc(-βπ)Γq(-β).Proof.
Interchanging the integral and summation by definition, the left hand side of (34) equals
(36)(qα+1;q)m(qβ+1;q)n(q;q)m(q;q)n∑k=0m‍∑j=0n‍(q-m;q)kq(k2)qk(m+α+1)(qα+1,q;q)k×(q-n;q)jq(j2)qj(n+β+1)(qβ+1,q;q)j∫0∞‍xβ+k+jdx(-x;q)∞=(qα+1;q)m(qβ+1;q)n(q;q)m(q;q)n∑k=0m‍∑j=0n‍(q-m;q)kq(k2)qk(m+α+1)(qα+1,q;q)k×(q-n;q)jq(j2)qj(n+β+1)(qβ+1,q;q)j×πcsc(-βπ)Γq(-β)(1-q)1+βq-jβ-(j+12)-k(β+j)-(k+12)×(qβ+1;q)j+k=πcsc(-βπ)Γq(-β)(1-q)1+β(qα+1;q)m(qβ+1;q)n(q;q)m(q;q)n×∑j=0n‍(q-n;q)jqjn(q;q)j∑k=0m‍(q-m,qβ+j+1;q)kqk(m+α-β-j)(qα+1,q;q)k=(-1)mq(α-β)m+(m2)πcsc(-βπ)Γq(-β)(1-q)1+β×(qβ-α-m+1;q)m(qβ+1;q)n(q;q)m(q;q)n×∑j=0n‍(q-n,qβ-α+1;q)j(q,qβ-α-m+1;q)jqj(n-m),
which is the right hand side of (34) by using the second formula of (23) and simplification. Similar to (34), the right hand side of (35) is equal to
(37)q-mα-nβ(qα+1;q)m(qβ+1;q)n(q;q)m(q;q)n×∑k=0m‍∑j=0n‍(q-m;q)k(q-n;q)j(qα+1,q;q)k(qβ+1,q;q)j×∫0∞‍xβ+k+j(x;q)∞dx=q-mα-nβ(qα-β;q)m(qβ+1;q)n(q;q)m(q;q)n(1-q)1+β×πcsc(-βπ)Γq(-β)q(β+1)mq(β+22)ϕ21[q-n,qβ-α+1qβ-α-m+1;q],
which is equivalent to the right hand side of (35) by using the first formula of (23) and simplification. The proof of Lemma 10 is complete.Lemma 11.
Iff(x) is a polynomial of degree m about x and is defined in the infinite interval (0,∞), which can be expanded in a series of the form
(38)f(x)=∑n=0m‍Cmnℒn(α)(x;q)orf(x)=∑n=0m‍Dmnℳn(α)(x;q),
where Cmn and Dmn are the nth Fourier-Laguerre coefficients, and both of them are independent of x, then one has
(39)Cmn=Γq(-α)(q;q)nqnΓ(α+1)Γ(-α)(qα+1;q)n(1-q)α+1×∫0∞xα(-x;q)∞f(x)ℒn(α)(x;q)dx,(40)Dmn=Γq(-α)(q;q)nqαn-n-(α+22)Γ(α+1)Γ(-α)(qα+1;q)n(1-q)α+1×∫0∞‍xα(x;q)∞f(x)ℳn(α)(x;q)dx.Proof.
Multiplying (38) by xαℒn(x;q)/(-x;q)∞ and integrating term by term over the interval (0,∞), using (6), we obtain the proof of (39). Similarly, taking β=α in (35), we deduce
(41)∫0∞‍ℳm(α)(x;q)ℳn(α)(x;q)xα(x;q)∞dx=-q(α+22)+n-αn(qα+1;q)n(q-α;q)∞(q;q)n(q;q)∞πδm,nsinπα,
so we also gain the proof of (40). The proof of Lemma 11 is complete.Lemma 12.
Formin{α,β}>-1, one has
(42)ℒn(α)(xy;q)=∑k=0n‍[n+αn-k]ykq(α-β)kℒk(β)(x;q)×ϕ21[qk-n,qβ+1+kqα+k+1;yqα-β+n-k],(43)ℳn(α)(xy;q)=∑k=0n‍[n+αn-k]ykq(k-n+β)k-αnℳk(β)(x;q)×ϕ21[qk-n,qβ+1+kqα+k+1;yq].Remark 13.
Replacingx by (1-q)x and letting q→1, we have [15, equation (11)]
(44)Ln(α)(xy)=∑k=0n‍(n+αn-k)ykLk(β)(x)F21×[-n+k,β+k+1α+k+1;y].
Setting α=β, (42) and (43) reduce to (29) and (30), respectively.Proof.
Let(45)A=Γq(-β)(q;q)kqkπcsc(-βπ)(qβ+1;q)k(1-q)1+β,B=(1-q)1+βπcsc(-βπ)Γq(-β).
By Lemmas 10 and 11, the coefficient of ℒn(α)(xy;q) expanded by ℒn(β)(x;q)(46)Cnk=A·∑j=0n‍(qα+1;q)n(y;q)n-jyj(q;q)n-j(qα+1;q)j×∫0∞‍ℒj(α)(x;q)ℒk(β)(x;q)xβ(-x;q)∞dx=AB·∑j=0n‍(qα+1;q)n(y;q)n-jyj(q;q)n-j(qα+1;q)j×(-1)k+jq(α-β)j+(j2)+(k2)×[β-αj-k][k+βk]q-kj=AB·∑j=0n-k‍(qα+1;q)n(y;q)n-j-kyj+k(q;q)n-j-k(qα+1;q)j+k×[β-αj][k+βk](-1)jq(α-β)(j+k)-k+(j2)=AB·(qα+1;q)n(y;q)n-kyk(q;q)n-k(qα+1;q)kq(α-β-1)k×[k+βk]ϕ32[qk-n,qα-β,0qk-n+1y,qα+k+1;q]=AB·(qα+1;q)nyk(q;q)n-k(qα+1;q)kq(α-β-1)k×[k+βk]ϕ21[qk-n,qβ+1+kqα+k+1;yqα-β+n-k]
is equal to the right hand side of (42). Similarly, we have
(47)Dnk=A·q-(β+22)-2k+βk×∫0∞‍ℳn(α)(xy;q)ℳk(β)(x;q)xβ(x;q)∞dx=AB·q-(β+22)-2k+βk×∑j=0n‍(-1)n-j(qα+1;q)n(1/y;q)n-jyn(q;q)n-j(qα+1;q)jq(j2)+αj-(n2)-αn×(-1)j+kq(j+12)+(k+12)-(α+j)k+(β+22)×[β-αj-k][k+βk]=AB·q-2k+βk-αn-(n2)+(k+12)-αk×∑j=0n-k‍(-1)n+k(qα+1;q)n(1/y;q)n-k(qn-j-k+1;q)jyn(qn-j-k/y;q)j(q;q)n-k(qα+1;q)j+k×q(j+k)2+α(j+k)-(j+k)k×(qβ-α-j+1;q)j(q;q)j[k+βk]=AB·q-2k+βk-αn-(n2)+(k+12)×(-1)n+k(qα+1;q)n(1/y;q)n-kyn(q;q)n-k(qα+1;q)k[k+βk]×ϕ22[qk-n,qα-βyqk+1-n,qα+k+1;yqβ+k+2],
which is equivalent to the right hand side of (43) by (25) and simplification. The proof of Lemma 12 is complete.Proof of Theorems2 and 3.
By using formula (42), the left hand side of (7) is equal to
(48)∑k=0m‍[m+αm-k]ykq(α-μ)k×ϕ21[qk-m,qμ+1+kqα+k+1;yqα-μ+m-k]×∑j=0n‍[n+βn-j]zjq(β-μ)j×ϕ21[qj-n,qμ+1+jqβ+j+1;zqβ-μ+n-j]×∫0∞‍ℒk(μ)(x;q)ℒj(μ)(x;q)xμ(-x;q)∞dx.
Similarly, with the help of formula (43), the left hand side of (8) equals
(49)∑k=0m‍[m+αm-k]ykq(k-m+μ)k-αm2ϕ1×[qk-m,qμ+1+kqα+k+1;yq]×∑j=0n‍[n+βn-j]zjq(j-n+μ)j-βn×ϕ21[qj-n,qμ+1+jqβ+j+1;zq]×∫0∞‍ℳk(μ)(x;q)ℳj(μ)(x;q)xμ(x;q)∞dx.
Using formulas (41) and (43) and noticing that the orthogonality of previous two types of q-Laguerre polynomials for the case of k=j, we can deduce (7) and (8). The proof of Theorems 2 and 3 is complete.


## 3. Representations ofq-Hermite Polynomials

Doha [34, page 5460] deduced the following result by third-order recurrence relation of the coefficients.Proposition 14 (see [34, equation (49)]).
Forα>-1 and n∈ℕ, one has
(50)Hn(x)=2n(1+α)n×∑k=0n‍F22[-(n-k)2,-(n-k-1)2-(α+n)2,-(α+n-1)2;-14]×(-n)kLk(α)(x)(1+α)k.In this section, we employ the technique of rearrangement of series(51)∑n=0∞‍∑k=0∞‍𝒜(k,n)=∑n=0∞‍∑k=0n‍𝒜(k,n-k)=∑n=0∞‍∑k=0[n/2]‍𝒜(k,n-2k)
to derive the following q-analogue of Proposition 14.Theorem 15.
Forα>-1 and n∈ℕ, one has
(52)ℋn(x;q)=(qα+1,q;q)n(1-q)-n/2∑k=0n‍C(n,k)ℒk(α)(x;q),
where
(53)C(n,k)=(-1)kq-(n2+n+4αn+4nk-2k2)/4(qα+1;q)k(q;q)n-k×∑s=0[(n-k)/2]‍(-1)s(qk-n;q)2s(1-q)s(q-(n+α);q)2s(-q-n/2;q1/2)2s(q;q)s,
and [x] denotes the greatest integer not exceeding x.Theorem 16.
Forα>-1 and n∈ℕ, one has
(54)𝒢n(x;q)=(qα+1,q;q)n(1-q)-n/2q-n(n-1)/4×∑k=0n‍D(n,k)ℳk(α)(x;q),
where
(55)D(n,k)=(-1)kq(k2)+αk(qα+1;q)k(q;q)n-k×∑s=0[(n-k)/2]‍(-1)sq(s2)(1-q)s(qk-n;q)2sq(2s2-s-4sk-4sα)/2(q;q)s(q-(n+α);q)2s(-q-n/2;q1/2)2s.Before the proof of Theorem15 the following lemma is necessary.Lemma 17.
Forα>-1 and n∈ℕ, one has
(56)xn=(qα+1,q;q)nq-(αn+n2)∑k=0n‍(-1)kq(n-k2)ℒk(α)(x;q)(qα+1;q)k(q;q)n-k,xn=(qα+1,q;q)nqn∑k=0n‍(-1)kq(k2)+αkℳk(α)(x;q)(q;q)n-k(qα+1;q)k.Proof.
Lettingf(x)=xm in (38) and using the following fact [8, page 23]:
(57)∫0∞‍ℒn(α)(x;q)xα+m(-x;q)∞dx=(q-m;q)n(qα+1;q)mΓ(-α)Γ(α+1)(q;q)nΓq(-α)qαm+(m+12)(1-q)-1-α,
similarly, we deduce the explicit representation of (39) and (40), respectively,
(58)Cmn=(qα+1;q)m(q-m;q)nqn(qα+1;q)nqαm+(α+12),Dmn=(qα+1;q)m(q-m;q)nqm(n+1)(qα+1;q)nq-αn,
so we obtain the formula (56). The proof is complete.Lemma 18 (see [7, equations (3.29.5) and (3.28.5)]).
One has(59)∑n=0∞‍tn(q1/2;q1/2)nℋn(x;q)=(-xt(1-q)-1/2;q1/2)∞(-t2;q)∞,∑n=0∞‍tnqn(n-1)/4(q1/2;q1/2)n𝒢n(x;q)=(t2;q)∞(xt(1-q)-1/2;q1/2)∞.Proof.
By using [7, equations (3.29.5) and (3.28.5)]
(60)∑n=0∞‍q(n2)(q;q)nh~n(x;q)tn=(-xt;q)∞(-t2;q2)∞,∑n=0∞‍hn(x;q)(q;q)ntn=(t2;q2)∞(xt;q)∞.
and replacing, respectively, by
(61)ℋn(x;q)=qn(n-1)/4h~n(x1-q;q1/2),𝒢n(x;q)=q-n(n-1)/4hn(x1-q;q1/2),
we deduce the proof of Lemma 18. The proof is complete.Proof of Theorem15.
From the generating function ofℋn(x;q), we have
(62)∑n=0∞‍tn(q1/2;q1/2)nℋn(x;q)=∑n=0∞‍qn(n-1)/4(xt)n(q1/2;q1/2)n(1-q)-n/2∑s=0∞‍(-t2)s(q;q)s=∑n,s=0∞‍qn(n-1)/4tn+2s(-1)sxn(q1/2;q1/2)n(q;q)s=∑n,s=0∞‍qn(n-1)/4tn+2s(-1)s(1-q)-n/2(q1/2;q1/2)n(q;q)s×(qα+1,q;q)nq-(αn+n2)×∑k=0n‍(-1)jq(n-k2)ℒk(α)(x;q)(qα+1;q)k(q;q)n-k=∑n,s,k=0∞‍[{12}({12}q(n+k)(n+k-1)/4(-1)stn+2s+k(qα+1;q)n+k×q-[α(n+k)+(n+k)2](1-q)-(n+k)/2{21})×((q1/2;q1/2)n+k(q;q)s)-1{22}]×(-1)kq(n2)ℒk(α)(x;q)(qα+1;q)k(q;q)n=∑n,k=0∞‍∑s=0[n/2]‍[{∏1}({22}q(n+k-2s)(n+k-2s-1)/4(-1)stn+k×(qα+1;q)n-2s+kq-[α(n-2s+k)+(n-2s+k)2]{21})×((q1/2;q1/2)n-2s+k(q;q)s)-1{22}]×(1-q)-(n+k-2s)/2(-1)kq(n-2s2)ℒk(α)(x;q)(qα+1;q)k(q;q)n-2s=∑n,k=0∞‍(-1)ktn+kℒk(α)(x;q)(1-q)-(n+k)/2(qα+1;q)k×∑s=0[n/2]‍(-1)s(qα+1,q;q)n-2s+k(1-q)s(q1/2;q1/2)n+k-2s(q;q)n-2s(q;q)s×q(n+k-2s)(n+k-2s-1)/4×q-[α(n+k-2s)+(n+k-2s)2]+(n-2s2)=∑n,k=0∞‍[{11}({21}(-1)ktn+kℒk(α)(x;q)(1-q)-(n+k)/2×(qα+1;q)n+k(-q1/2;q1/2)n+k{11})×((qα+1;q)k(q;q)n)-1{12}]×∑s=0[n/2]‍(-1)s(q-n;q)2s(1-q)s(q-(n+k+α);q)2s(-q-(n+k)/2;q1/2)2s(q;q)s×q(n+k-2s)(n+k-2s-1)/4-[α(n+k-2s)+(n+k-2s)2]×q+(n-2s2)-2s(α+k)-s(n+k)+s(2s-1)/2=∑n=0∞‍(qα+1;q)n(-q1/2;q1/2)n(1-q)-n/2tn×∑k=0n‍(-1)kℒk(α)(x;q)(qα+1;q)k(q;q)n-k×∑s=0[(n-k)/2]‍(-1)s(qk-n;q)2s(1-q)s(q-(n+α);q)2s(-q-n/2;q1/2)2s(q;q)s×q(n-2s)(n-2s-1)/4-[α(n-2s)+(n-2s)2]×q+(n-k-2s2)-2s(α+k)-sn+s(2s-1)/2.
Comparing the coefficients of tn on both sides of (62), we obtain the results.Proof of Theorem16.
From the generating function of𝒢n(x;q), we have
(63)∑n=0∞‍tnqn(n-1)/4(q1/2;q1/2)n𝒢n(x;q)=∑n,s=0∞‍(-1)sq(s2)xn(1-q)-n/2tn+2s(q1/2;q1/2)n(q;q)s=∑n,s,k=0∞[({11}(-1)sq(s2)+n(1-q)-(n+k)/2tn+k+2s×(qα+1;q)n+k(-1)kq(k2)+αkℳk(α)(x;q){11})×((q1/2;q1/2)n+k(q;q)n(qα+1;q)k(q;q)s)-1{11}]=∑n,k=0∞‍(-1)kq(k2)+αk+nℳk(α)(x;q)(1-q)-(n+k)/2tn+k(qα+1;q)k×∑s=0[n/2]‍(-1)sq(s2)-2s(1-q)s(qα+1;q)n-2s+k(q1/2;q1/2)n+k-2s(q;q)n-2s(q;q)s=∑n,k=0∞‍(-1)kq(k2)+αk+nℳk(α)(x;q)(1-q)-(n+k)/2tn+k(qα+1;q)k×(qα+1;q)n+k(-q1/2;q1/2)n+k(q;q)n×∑s=0[n/2]‍(-1)sq(s2)-2s(1-q)s(q-n;q)2s(q;q)s(q-(n+k+α);q)2s(-q-(n+k)/2;q1/2)2s×q-2s(k+α)+s(2s-1)/2=∑n=0∞‍(qα+1;q)n(-q1/2;q1/2)n(1-q)-n/2tn×∑k=0n‍(-1)kq(k2)+αk+n-kℳk(α)(x;q)(qα+1;q)k(q;q)n-k×∑s=0[(n-k)/2]‍(-1)sq(s2)-2s(1-q)s(qk-n;q)2s(q;q)s(q-(n+α);q)2s(-q-n/2;q1/2)2s×q-2s(k+α)+s(2s-1)/2.
Equating the coefficients of tn on both sides of (63), we obtain the results.Proof of Corollary23.
In view of the fact that(64)(qβ;q)2n=(qβ/2,-qβ/2,q(β+1)/2,-q(β+1)/2;q)n,|β|∈ℕ,
we have
(65)limq→1∑s=0[(n-k)/2]‍(-1)s(qk-n;q)2s(1-q)sq(s2)-2s(q-(n+α);q)2s(-q-n/2;q1/2)2s(q;q)s=limq→1∑s=0[(n-k)/2][q-(n+α-1)/2,-q-(n+α-1)/2;q)S)-1(-q-(n-k-1)/2;q)S(-1)s(q-(n-k)/2,-q-(n-k)/2,q-(n-k-1)/2,-q-(n-k-1)/2;q)S)×((q-(n+α)/2,-q-(n+α)/2,q-(n+α-1)/2,-q-(n+α-1)/2;q)S)-1]×(1-q)sq(s2)-2s(-q-n/2;q1/2)2s(q;q)s=F22[-(n-k)2,-(n-k-1)2-(α+n)2,-(α+n-1)2;-14],limx→(1-q)x,q→1(qα+1;q)n(q;q)n(1-q)n∑k=0n‍(-1)kℒk(α)(x;q)(qα+1;q)k(q;q)n-k×q-n2/4-n/4-αn-nk+k2/2=limx→(1-q)x,q→1(qα+1,q;q)n(1-q)nq-n(n-1)/4×∑k=0n‍(-1)kq(k2)+αkℳk(α)(x;q)(qα+1;q)k(q;q)n-k=(1+α)n∑k=0n‍(-n)kLk(α)(x)(1+α)k.
Combing (65) and (2), we deduce (50). The proof of Corollary 23 is complete.


## 4. Integrals Related to Generalizedq-Hermite Polynomials

The authors [23] deduced the following interesting result inspired by the relation of (16) and the orthogonality of q-Laguerre polynomials (6).Proposition 19 (see [23, Theorem 1]).
The sequence of theq-polynomials {ℋn(μ)(x;q)}, which are defined by the relations (16), satisfies the orthogonality relation
(66)∫-∞∞‍ℋm(μ)(x;q)ℋn(μ)(x;q)|x|2μ(-x2;q)∞dx=πcosπμ(q1/2-μ;q)∞(q;q)∞q-n/2-μθn×(q;q)[n/2](qμ+1/2;q)[(n+1)/2]δmn
on the whole real line ℝ, where θn=n-2[n/2].In this section, we will further consider multivariateq-Hermite polynomials by Theorems 2 and 3.Theorem 20.
Forθn=n-2[n/2], one has
(67)∫-∞∞‍ℋm(α)(xy;q)ℋn(β)(xz;q)|x|2μ(-x2;q)∞dx=(-1)[m/2]+[n/2](q;q)[m/2](q;q)[n/2]×(1-q)μ+1/2+θnπcsc(-μ+1/2-θn)πΓq(-μ+1/2-θn)×∑k=0min{[m/2],[n/2]}‍q(α+β-2μ-1)k(yz)2k+θn×ϕ21[qk-[m/2],qμ+1/2+k+θnqα+k+1/2+θn;y2qα-μ+[m/2]-k]×ϕ21[qk-[n/2],qμ+1/2+k+θnqβ+k+1/2+θn;z2qβ-μ+[n/2]-k]×[[m2]+α-12+θn[m2]-k][[n2]+β-12+θn[n2]-k]×[k+μ-12+θnk].Theorem 21.
Forθn=n-2[n/2], one has
(68)∫-∞∞‍𝒢m(α)(xy;q)𝒢n(β)(xz;q)|x|2μ(x2;q)∞dx=(q;q)[m/2](q;q)[n/2](1-q)μ+1/2+θn×q(μ+3/2+θn2)-(α-1/2+θn)[m/2]-(β-1/2+θn)[n/2]×πcsc(-μ+1/2-θn)πΓq(-μ+1/2-θn)×∑k=0min{[m/2],[n/2]}‍q(2k+μ+1/2-[m/2]-[n/2]+θn)k×(yz)2k+θn2ϕ1[qk-[m/2],qμ+1/2+k+θnqα+k+1/2+θn;y2q]×ϕ21[qk-[n/2],qμ+1/2+k+θnqβ+k+1/2+θn;z2q]×[[m2]+α-12+θn[m2]-k][[n2]+β-12+θn[n2]-k]×[k+μ-12+θnk].Remark 22.
Forα=β=μ and y=z=1, Theorems 20 and 21 reduce to Proposition 19 and Corollary 23 respectively.Corollary 23.
The sequence of theq-polynomials {𝒢n(μ)(x;q)}, which are defined by the relations (16), satisfies the orthogonality relation
(69)∫-∞∞‍𝒢m(μ)(x;q)𝒢n(μ)(x;q)|x|2μ(-x2;q)∞dx=πcosπμ(q1/2-μ;q)∞(q;q)∞×q(μ+3/22)-[n/2](μ-3/2+θn)+θn×(q;q)[n/2](qμ+1/2;q)[(n+1)/2]δmn
on the whole real line ℝ, where θn=n-2[n/2].Proof of Theorems20 and 21.
Letℐ(α,y,m;β,z,n;μ) and 𝒥(α,y,m;β,z,n;μ) represent the right hand side of (7) and (8) respectively. Since the weight function in (67) is an even function of the independent variable x by the definition (16), so the polynomials are evidently orthogonal to each other when the degrees m and n are either simultaneously even or odd. From (16) and Theorem 2 we have
(70)∫-∞∞‍ℋ2m(α)(xy;q)ℋ2n(β)(xz;q)|x|2μ(-x2;q)∞dx=(-1)m+n(q;q)m(q;q)n×∫-∞∞‍ℒm(α-1/2)(x2y2;q)ℒn(β-1/2)×(x2z2;q)|x|2μ(-x2;q)∞dx=2(-1)m+n(q;q)m(q;q)n×∫0∞‍ℒm(α-1/2)(x2y2;q)ℒn(β-1/2)×(x2z2;q)|x|2μ(-x2;q)∞dx=(-1)m+n(q;q)m(q;q)n×∫0∞‍ℒm(α-1/2)(y2t;q)ℒn(β-1/2)(z2t;q)tμ-1/2(-t;q)∞dt=(-1)m+n(q;q)m(q;q)n×ℐ(α-12,y2,m;β-12,z2,n;μ-12),(71)∫-∞∞‍ℋ2m+1(α)(xy;q)ℋ2n+1(β)(xz;q)|x|2μ(-x2;q)∞dx=2(-1)m+nyz(q;q)m(q;q)n×∫0∞‍ℒm(α+1/2)(x2y2;q)ℒn(β+1/2)(x2z2;q)×|x|2μ+2(-x2;q)∞dx=(-1)m+nyz(q;q)m(q;q)n×∫0∞‍ℒm(α+1/2)(y2t;q)ℒn(β+1/2)(z2t;q)tμ+1/2(-t;q)∞dt=(-1)m+nyz(q;q)m(q;q)n×ℐ(α+12,y2,m;β+12,z2,n;μ+12).
Putting (70) and (71) together and using Theorem 2, we complete the proof of Theorem 20 after some simplification. In the same way we find that
(72)∫-∞∞‍𝒢2m(α)(xy;q)𝒢2n(β)(xz;q)|x|2μ(-x2;q)∞dx=q-(m+12)-(n+12)(q;q)m(q;q)n𝒥×(α-12,y2,m;β-12,z2,n;μ-12),∫-∞∞‍𝒢2m+1(α)(xy;q)𝒢2n+1(β)(xz;q)|x|2μ(-x2;q)∞dx=q-(m+12)-(n+12)yz(q;q)m(q;q)n×𝒥(α+12,y2,m;β+12,z2,n;μ+12),
which are two cases of Theorem 21; thus we obtain the proof.Proof of Proposition19 and Corollary 23.
Let us consider first that the case of bothm and n is even, and just take α=β=μ and y=z=1 in Theorem 20. We have
(73)∫-∞∞‍ℋ2m(μ)(x;q)ℋ2n(μ)(x;q)|x|2μ(-x2;q)∞dx=(-1)m+n(q;q)m(q;q)n(1-q)μ+1/2×πcsc(-μ+1/2)πΓq(-μ+1/2)∑k=0min{m,n}‍q-k×ϕ10[qk-m-;qm-k]ϕ10[qk-n-;qn-k]×[m+α-12m-k][n+β-12n-k][k+μ-12k]=(q;q)n(q;q)n(1-q)μ+1/2×πcsc(-μ+1/2)πΓq(-μ+1/2)q-n[n+μ-12n]δmn=πcosμπ(q1/2-μ;q)∞(q;q)∞q-n(q;q)n(qμ+1/2;q)nδmn,(74)∫-∞∞‍ℋ2m+1(μ)(x;q)ℋ2n+1(μ)(x;q)|x|2μ(-x2;q)∞dx=(q;q)n(q;q)n(1-q)μ+3/2×πcsc(-μ-1/2)πΓq(-μ-1/2)q-n[n+μ+12n]δmn=-πcosμπ(q-μ-1/2;q)∞(q;q)∞×q-n(q;q)n(q;q)n+μ+1/2(q;q)μ+1/2δmn=πcosμπ(q1/2-μ;q)∞(q;q)∞×q-n-μ-1/2(q;q)n(qμ+1/2;q)n+1δmn.
Putting (73) and (74) together results in the orthogonality relation (66). The proof of Proposition 19 is complete. By the same way, we can deduce Corollary 23. This completes the proof.



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*Source: 302642-2013-07-17.xml*