Some Integrals Involving 𝑞-Laguerre Polynomials and Applications

Abstract

The integrals involving multivariate q-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.

Dedicated to Srinivasa Ramanujan on the occasion of his 125th birth anniversary

1. Introduction

The 𝑞-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 𝑠𝑢_𝑞(1∣𝑞)-representation by [3]. For more information, please refer to [1–5].

The 𝑞-Laguerre polynomials are defined by [6, equation (1.0.1)] ℒ_𝑛^(𝛼)=𝑞𝑥;𝑞^𝛼+1;𝑞_𝑛𝑞;𝑞_𝑛^𝑛_𝑘=0𝑞^−𝑛;𝑞_𝑘𝑞^𝑘2𝑥𝑞^{𝑛+𝛼+1𝑘}𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑘, (1) 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 auxiliary 𝑞-Laguerre polynomials as follows: ℳ_𝑛^(𝛼)𝑥;𝑞=𝑞^−𝑛𝛼𝑞^𝛼+1;𝑞_𝑛𝑞;𝑞_𝑛^𝑛_𝑘=0𝑞^−𝑛;𝑞_𝑘𝑥^𝑘𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑘. (2)

It is easy to see the validity of the following: ℒ_𝑛^(𝛼)1−𝑞𝑥;𝑞=ℳ_𝑛^(𝛼)1−𝑞⁻¹𝑥;𝑞⁻¹,lim_𝑞→1−ℒ_𝑛^(𝛼)1−𝑞𝑥;𝑞=lim_𝑞→1+ℳ_𝑛^(𝛼)1−𝑞𝑥;𝑞=𝐿_𝑛^(𝛼)(𝑥), (3)(4) where the classical Laguerre polynomials are defined by [9, page 201] 𝐿_𝑛^(𝛼)(𝑥)=^𝑛_𝑘=0(−1)^𝑘(1+𝛼)_𝑛𝑥^𝑘𝑘!(𝑛−𝑘)!(1+𝛼)_𝑘. (5) For more information about classical Laguerre polynomials, please refer to [9–15] and the references therein.

The well-known orthogonality of 𝑞-Laguerre polynomials reads the following.

Proposition 1 (see [6, equation (2.0.7)] and [8, equation (2.4)]). For 𝛼>−1 and for 𝑚,𝑛∈ℕ, one has ^∞₀ℒ_𝑚^(𝛼)𝑥;𝑞ℒ_𝑛^(𝛼)𝑥𝑥;𝑞^𝛼−𝑥;𝑞_∞d𝑥=−𝑞^−𝑛𝑞^𝛼+1;𝑞_𝑛𝑞^−𝛼;𝑞_∞𝑞;𝑞_𝑛𝑞;𝑞_∞𝜋𝛿_𝑚,𝑛.sin𝜋𝛼 (6)

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 Proposition 1 and the auxiliary ones as follows.

Theorem 2. For ℜ{𝜇}>−1 and 𝑚,𝑛∈ℕ, one has ^∞₀ℒ_𝑚^(𝛼)𝑥𝑦;𝑞ℒ_𝑛^(𝛽)𝑥𝑥𝑧;𝑞^𝜇−𝑥;𝑞_∞=𝑑𝑥1−𝑞^1+𝜇𝜋csc−𝜇𝜋Γ_𝑞×−𝜇_{min{𝑚,𝑛}𝑘=0}𝑘𝑞𝑚+𝛼𝑚−𝑘𝑛+𝛽𝑛−𝑘𝑘+𝜇_{(𝛼+𝛽−2𝜇−1)𝑘}𝑦^𝑘𝑧^𝑘×₂𝜙₁𝑞^𝑘−𝑚,𝑞^𝜇+1+𝑘𝑞^𝛼+𝑘+1;𝑦𝑞^{𝛼−𝜇+𝑚−𝑘}×₂𝜙₁𝑞^𝑘−𝑛,𝑞^𝜇+1+𝑘𝑞^𝛽+𝑘+1;𝑧𝑞^{𝛽−𝜇+𝑛−𝑘}. (7)

Theorem 3. For ℜ{𝜇}>−1 and 𝑚,𝑛∈ℕ, one has ^∞₀ℳ_𝑚^(𝛼)𝑥𝑦;𝑞ℳ_𝑛^(𝛽)𝑥𝑧;𝑞𝑥^𝜇𝑥;𝑞_∞𝑑𝑥=𝑞^{2𝜇+2−𝛼𝑚−𝛽𝑛}1−𝑞^1+𝜇𝜋csc−𝜇𝜋Γ_𝑞×−𝜇_{min{𝑚,𝑛}𝑘=0}𝑘𝑚+𝛼𝑚−𝑘𝑛+𝛽𝑛−𝑘𝑘+𝜇×𝑞^{(2𝑘+𝜇−𝑚−𝑛+1)𝑘}𝑦^𝑘𝑧^𝑘×₂𝜙₁𝑞^𝑘−𝑚,𝑞^𝜇+1+𝑘𝑞^𝛼+𝑘+1;𝑦𝑞₂𝜙₁𝑞^𝑘−𝑛,𝑞^𝜇+1+𝑘𝑞^𝛽+𝑘+1;𝑧𝑞. (8)

Corollary 4 (see [15, equation (14)]). For ℜ{𝛾}>−1, ℜ{𝜎}>0, and 𝑚,𝑛∈ℕ, one has ^∞₀𝑥^𝛾𝑒^−𝜎𝑥𝐿_𝑚^(𝛼)(𝜆𝑥)𝐿_𝑛^(𝛽)=Γ𝜇𝑥𝑑𝑥𝛾+1𝜎^{𝛾+1min{𝑚,𝑛}}_𝑘=0×𝑘𝑚+𝛼𝑚−𝑘𝑛+𝛽𝑛−𝑘𝑘+𝛾𝜆𝜇𝜎^2𝑘×₂𝐹₁;−𝑚+𝑘,𝛾+𝑘+1𝜆𝛼+𝑘+1𝜎×₂𝐹₁;−𝑛+𝑘,𝛾+𝑘+1𝜇𝛽+𝑘+1𝜎. (9)

Remark 5. Theorems 2 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 discrete 𝑞-Hermite polynomials ℎ_𝑛(𝑥;𝑞) and ℎ_𝑛(𝑥;𝑞) are defined by [7, pages 90-91] ℎ_𝑛𝑥;𝑞=𝑥^𝑛₂𝜙₀𝑞^−𝑛,𝑞^−𝑛+1−;𝑞²,𝑞^2𝑛−1𝑥²≜𝑞^𝑛2𝒢_𝑛𝑥1−𝑞²;𝑞²,ℎ_𝑛𝑥;𝑞=𝑥^𝑛₂𝜙₁𝑞^−𝑛,𝑞^−𝑛+10;𝑞²,𝑞²𝑥²≜𝑞^−𝑛2ℋ_𝑛𝑥1−𝑞²;𝑞², (10) which are equivalent to Al-Salam-Carlitz polynomials with 𝑎=−1 (please refer to [22, page 53] also), and the relation between them is ℎ_𝑛(𝑖𝑥;𝑞⁻¹)=𝑖^𝑛ℎ_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 new 𝑞-Hermite polynomials ℋ_𝑛(𝑥;𝑞) and 𝒢_𝑛(𝑥;𝑞), whose names come from the facts lim_𝑞→11−𝑞^−𝑛/2ℋ_𝑛1−𝑞𝑥;𝑞=lim_𝑞→11−𝑞^−𝑛/2𝒢_𝑛1−𝑞𝑥;𝑞=2^−𝑛𝐻_𝑛(𝑥),lim_{𝑥→1−𝑞𝑥,𝑡→}^∞_{1−𝑞𝑡,𝑞→1𝑛=0}𝑡^𝑛𝑞^1/2;𝑞^1/2_𝑛ℋ_𝑛𝑥;𝑞=exp2𝑥𝑡−𝑡²=^∞_𝑛=0𝑡^𝑛𝐻𝑛!^𝑛(𝑥)=lim_{𝑥→1−𝑞𝑥,𝑡→}^∞_{1−𝑞𝑡,𝑞→1𝑛=0}𝑡^𝑛𝑞^{𝑛(𝑛−1)/4}𝑞^1/2;𝑞^1/2_𝑛𝒢_𝑛𝑥;𝑞, (11)(12) then we deduce the representations of ℋ_𝑛(𝑥;𝑞) and 𝒢_𝑛(𝑥;𝑞) by 𝑞-Laguerre polynomials; see Theorems 15 and 16.

As an application, using the orthogonality of 𝑞-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. For 𝛼>−1 and 𝑗≤𝑛∈ℕ, one has ^∞₀ℋ_𝑛𝑥;𝑞ℒ_𝑗^(𝛼)𝑥𝑥;𝑞^𝛼−𝑥;𝑞_∞𝑞𝑑𝑥=−^𝛼+1,𝑞;𝑞_𝑛𝑞^𝛼+1;𝑞_𝑗𝑞^−𝛼;𝑞_∞1−𝑞^−𝑛/2𝑞;𝑞_𝑗𝑞;𝑞_∞×𝜋𝐶𝑛,𝑗,sin𝜋𝛼 (13) where 𝐶(𝑛,𝑘) is defined by (52).

Theorem 7. For 𝛼>−1 and 𝑗≤𝑛∈ℕ, one has ^∞₀𝒢_𝑛𝑥;𝑞ℳ_𝑗^(𝛼)𝑥;𝑞𝑥^𝛼𝑥;𝑞_∞𝑑𝑥=−𝑞^{𝑛2+2−(1/2)𝛼+2+𝑗−𝛼𝑗}×𝑞^𝛼+1,𝑞;𝑞_𝑛𝑞^𝛼+1;𝑞_𝑗𝑞^−𝛼;𝑞_∞1−𝑞^−𝑛/2𝑞;𝑞_𝑗𝑞;𝑞_∞×𝜋𝐷𝑛,𝑗,sin𝜋𝛼 (14) 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: 𝐻^(𝜇)_2𝑛(𝑥)=(−1)^𝑛2^2𝑛𝑛!𝐿_𝑛^(𝜇−1/2)𝑥²,𝐻^(𝜇)_2𝑛+1(𝑥)=(−1)^𝑛2^2𝑛+1𝑛!𝑥𝐿_𝑛^(𝜇+1/2)𝑥². (15)

The authors [23, equation (2.7)] defined the following generalized 𝑞-Hermite polynomials: ℋ^(𝜇)_2𝑛𝑥;𝑞=(−1)^𝑛𝑞;𝑞_𝑛ℒ_𝑛^(𝜇−1/2)𝑥²,ℋ;𝑞_{(𝜇)2𝑛+1}𝑥;𝑞=(−1)^𝑛𝑞;𝑞_𝑛𝑥ℒ_𝑛^(𝜇+1/2)𝑥²;𝑞 (16) and deduced their orthogonal relations; see Proposition 19 below.

In this paper, we continue to define the auxiliary polynomials according to (16) as follows: 𝒢^(𝜇)_2𝑛𝑥;𝑞=𝑞^−2𝑛+1𝑞;𝑞_𝑛ℳ_𝑛^(𝜇−1/2)𝑥²,𝒢;𝑞_{(𝜇)2𝑛+1}𝑥;𝑞=𝑞^−2𝑛+1𝑞;𝑞_𝑛𝑥ℳ_𝑛^(𝜇+1/2)𝑥²;𝑞. (17)

With the aid of (15)–(17) and (4), one readily verifies that lim_𝑞→11−𝑞^−𝑛/2ℋ_𝑛^(𝜇)1−𝑞𝑥;𝑞=lim_𝑞→11−𝑞^−𝑛/2𝒢_𝑛^(𝜇)1−𝑞𝑥;𝑞=2^−𝑛𝐻_𝑛^(𝜇)(𝑥). (18)

As another application of this paper, we gain the general 𝑞-Laguerre polynomials of several variables by Theorems 2 and 3, and we also deduce the orthogonal polynomials of 𝒢_𝑛^(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 Section 2, 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 Theorems 2 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 𝑎;𝑞₀=1,𝑎;𝑞_𝑛=^𝑛−1_𝑘=01−𝑎𝑞^𝑘,𝑎;𝑞_∞=^∞_𝑘=01−𝑎𝑞^𝑘, (19) and (𝑎₁,𝑎₂,…,𝑎_𝑚;𝑞)_𝑛=(𝑎₁;𝑞)_𝑛(𝑎₂;𝑞)_𝑛⋯(𝑎_𝑚;𝑞)_𝑛, where 𝑚 is a positive integer and 𝑛 is a nonnegative integer or ∞.

The basic hypergeometric series _𝑟𝜙_𝑠 is given by _𝑟𝜙_𝑠𝑎₁,…,𝑎_𝑟𝑏₁,…,𝑏_𝑠=;𝑞,𝑧_∞𝑛=0𝑎₁,𝑎₂,…,𝑎_𝑟;𝑞_𝑛𝑞,𝑏₁,…,𝑏_𝑠;𝑞_𝑛𝑧^𝑛(−1)^𝑛𝑞^{𝑛2𝑠+1−𝑟}. (20) 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 (𝑎_𝑖,𝑏_𝑖)=(𝑞^𝑎𝑖,𝑞^𝑏𝑖) and setting 𝑞→1, (20) reduces to the classical Gauss’ hypergeometric series _𝑟𝐹_𝑠𝑎₁,𝑎₂,…,𝑎_𝑟𝑏₁,…,𝑏_𝑠;𝑧=^∞_𝑛=0𝑎_1𝑛𝑎_2𝑛⋯𝑎_𝑟𝑛𝑛!𝑏_1𝑛⋯𝑏_𝑠𝑛𝑧^𝑛, (21) where Pochhammer symbol (𝑧)_𝑛 is defined by (𝑧)_𝑛=𝑧(𝑧+1)⋯(𝑧+𝑛−1)=Γ(𝑧+𝑛)/Γ(𝑧).

The 𝑞-analogue of the gamma function is defined by (see [32, equation (1.10.1)]) as follows: Γ_𝑞(𝑥)=𝑞;𝑞_∞𝑞^𝑥;𝑞_∞1−𝑞^1−𝑥,0<𝑞<1. (22)

The 𝑞-Chu-Vandermonde formula [32, equations (II.6) and (II.7)] reads that ₂𝜙₁𝑞^−𝑛𝑐=,𝑏;𝑞,𝑞𝑐/𝑏;𝑞_𝑛𝑐;𝑞_𝑛𝑏^𝑛,₂𝜙₁𝑞^−𝑛𝑐,𝑏;𝑞,𝑐𝑞^𝑛𝑏=𝑐/𝑏;𝑞_𝑛𝑐;𝑞_𝑛. (23)

The ₃𝜙₂ transformations [32, equations (III.12) and (III.13)] stats that ₃𝜙₂𝑞^−𝑛=,𝑎,𝑐𝑎𝑏,𝑑;𝑞,𝑞^𝑛𝑑/𝑎;𝑞_𝑛𝑑;𝑞_𝑛×₃𝜙₂𝑞^−𝑛𝑏,𝑎,𝑐𝑏,𝑎𝑞^1−𝑛𝑑;𝑞,𝑐𝑞𝑑₃,𝜙₂𝑞^−𝑛,𝑏,𝑐𝑑,𝑒;𝑞,𝑑𝑒𝑞^𝑛=𝑏𝑐𝑒/𝑐;𝑞_𝑛𝑒;𝑞_𝑛×₃𝜙₂𝑞^−𝑛𝑑,𝑐,𝑏𝑑,𝑐𝑞^1−𝑛𝑒;𝑞,𝑐𝑞𝑑. (24)

The 𝑞-analogue of the Pfaff-Kummer transformation [32, equation (III.4)] is as follows: ₂𝜙₁𝑐=𝑎,𝑏;𝑞,𝑧𝑎𝑧;𝑞_∞𝑧;𝑞_∞²𝜙₂𝑐𝑎,𝑏.𝑐,𝑎𝑧;𝑞,𝑏𝑧 (25)

The Ramanujan beta integral is stated as follows [33, equation (2.8)]: ^∞₀𝑡^𝑥−1−𝑎𝑡;𝑞_∞−𝑡;𝑞_∞𝑑𝑡=𝑎,𝑞^1−𝑥;𝑞_∞𝑞,𝑎𝑞^−𝑥;𝑞_∞𝜋,sin𝜋𝑥0<𝑎<𝑞^𝑥,𝑥>0. (26)

Lemma 8 (see [33, equation (4.2)]). One has ^∞₀𝑥^𝛼−𝑥;𝑞_∞Γ()Γ()𝑑𝑥=−𝛼𝛼+11−𝑞^1+𝛼Γ_𝑞()𝑞−𝛼=−^−𝛼;𝑞_∞𝑞;𝑞_∞𝜋,sin𝜋𝛼_∞0𝑥^𝛼𝑥;𝑞_∞𝑑𝑥=−𝑞^2𝛼+2𝑞^−𝛼;𝑞_∞𝑞;𝑞_∞𝜋.sin𝜋𝛼 (27)(28)

Proof. Taking (𝑎,𝑡)=(−1/𝑎,𝑎𝑡) in (26), then letting 𝑎→0, we obtain (28) immediately. The proof is complete.

Lemma 9. For 𝛼>−1 and 𝑛∈ℕ, one has ℒ_𝑛^(𝛼)𝑥𝑦;𝑞=^𝑛_𝑘=0𝑞^𝛼+1;𝑞_𝑛𝑦;𝑞_𝑛−𝑘𝑦^𝑘ℒ_𝑘^(𝛼)𝑥;𝑞𝑞;𝑞_𝑛−𝑘𝑞^𝛼+1;𝑞_𝑘,ℳ_𝑛^(𝛼)=𝑥𝑦;𝑞_𝑛𝑘=0(−1)^𝑛−𝑘𝑞^𝛼+1;𝑞_𝑛1/𝑦;𝑞_𝑛−𝑘𝑦^𝑛ℳ_𝑘^(𝛼)𝑥;𝑞𝑞;𝑞_𝑛−𝑘𝑞^𝛼+1;𝑞_𝑘×𝑞^{𝑘2𝑛2+𝛼𝑘−−𝛼𝑛}. (29)(30)

Proof. Letting 𝛾=0 in [6, Proposition 4.1], ^∞_𝑛=0𝛾𝑡;𝑞_𝑛𝑞^{𝛼𝑛+𝑛2}𝛾𝑡,𝑞^𝛼+1,𝑞;𝑞_𝑛(−𝑥𝑡)^𝑛=𝑡;𝑞_∞𝛾𝑡;𝑞_∞^∞_𝑛∑₌₀𝛾;𝑞_𝑛𝑡^𝑛𝑞^𝛼+1;𝑞_𝑛ℒ_𝑛^(𝛼)𝑥;𝑞, (31) then replacing 𝑥 by 𝑥𝑦, we have ^∞_𝑛=0ℒ_𝑛^(𝛼)𝑥𝑦;𝑞𝑞^𝛼+1;𝑞_𝑛𝑡^𝑛=𝑦𝑡;𝑞_∞𝑡;𝑞_∞1𝑦𝑡;𝑞_∞^∞_𝑛=0𝑞^{𝛼𝑛+𝑛2}𝑞^𝛼+1,𝑞;𝑞_𝑛−𝑥𝑦𝑡^𝑛=^∞_𝑘=0𝑦;𝑞_𝑘𝑞;𝑞_𝑘𝑡^𝑘∞_𝑛=𝑘𝐿^(𝛼)_𝑛−𝑘𝑥;𝑞𝑦𝑡^𝑛−𝑘𝑞^𝛼+1;𝑞_𝑛−k. (32) Comparing the coefficients of 𝑡^𝑛 on both sides of (32) yields (29). Similar to (32), by the definition (1), we have ^∞_𝑛=0𝛾;𝑞_𝑛𝑡𝑞^𝛼𝑛𝑞^𝛼+1;𝑞_𝑛ℳ_𝑛^(𝛼)=𝑥;𝑞𝛾𝑡;𝑞_∞𝑡;𝑞_∞^∞_𝑛=0𝛾;𝑞_𝑛𝑥^𝑛𝑞/𝑡,𝑞^𝛼+1,𝑞;𝑞_𝑛. (33) 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. For min{𝛼,𝛽}>−1 and 𝑚,𝑛∈ℕ, one has ^∞₀ℒ_𝑚^(𝛼)𝑥;𝑞ℒ_𝑛^(𝛽)𝑥𝑥;𝑞^𝛽−𝑥;𝑞_∞=()𝑑𝑥−1^𝑚+𝑛𝑞^{𝑚2+𝑛2(𝛼−𝛽)𝑚+−𝑚𝑛}×1−𝑞^1+𝛽𝑛𝛽−𝛼𝜋𝑚−𝑛𝑛+𝛽csc−𝛽𝜋Γ_𝑞,−𝛽_∞0ℳ_𝑗^(𝛼)𝑥;𝑞ℳ_𝑘^(𝛽)𝑥;𝑞𝑥^𝛽𝑥;𝑞_∞=()𝑑𝑥−1^𝑘+𝑗𝑞^{2+22𝑗+1𝑘+1−(𝛼+𝑗)𝑘+𝛽+2}×1−𝑞^1+𝛽𝑘𝛽−𝛼𝜋𝑗−𝑘𝑘+𝛽csc−𝛽𝜋Γ_𝑞.−𝛽 (34)(35)

Proof. Interchanging the integral and summation by definition, the left hand side of (34) equals 𝑞^𝛼+1;𝑞_𝑚𝑞^𝛽+1;𝑞_𝑛𝑞;𝑞_𝑚𝑞;𝑞_𝑛^𝑚𝑛_𝑘=0𝑗=0𝑞^−𝑚;𝑞_𝑘𝑞^𝑘2𝑞^{𝑘(𝑚+𝛼+1)}𝑞^𝛼+1,𝑞;𝑞_𝑘×𝑞^−𝑛;𝑞_𝑗𝑞^𝑗2𝑞^{𝑗(𝑛+𝛽+1)}𝑞^𝛽+1,𝑞;𝑞_𝑗^∞₀𝑥^{𝛽+𝑘+𝑗}𝑑𝑥−𝑥;𝑞_∞=𝑞^𝛼+1;𝑞_𝑚𝑞^𝛽+1;𝑞_𝑛𝑞;𝑞_𝑚𝑞;𝑞_𝑛^𝑚𝑛_𝑘=0𝑗=0𝑞^−𝑚;𝑞_𝑘𝑞^𝑘2𝑞^{𝑘(𝑚+𝛼+1)}𝑞^𝛼+1,𝑞;𝑞_𝑘×𝑞^−𝑛;𝑞_𝑗𝑞^𝑗2𝑞^{𝑗(𝑛+𝛽+1)}𝑞^𝛽+1,𝑞;𝑞_𝑗×𝜋csc−𝛽𝜋Γ_𝑞−𝛽1−𝑞^1+𝛽𝑞^{22−𝑗𝛽−𝑗+1−𝑘(𝛽+𝑗)−𝑘+1}×𝑞^𝛽+1;𝑞_𝑗+𝑘=𝜋csc−𝛽𝜋Γ_𝑞−𝛽1−𝑞^1+𝛽𝑞^𝛼+1;𝑞_𝑚𝑞^𝛽+1;𝑞_𝑛𝑞;𝑞_𝑚𝑞;𝑞_𝑛×^𝑛_𝑗=0𝑞^−𝑛;𝑞_𝑗𝑞^𝑗𝑛𝑞;𝑞_𝑗^𝑚_𝑘=0𝑞^−𝑚,𝑞^𝛽+𝑗+1;𝑞_𝑘𝑞^{𝑘(𝑚+𝛼−𝛽−𝑗)}𝑞^𝛼+1,𝑞;𝑞_𝑘=(−1)^𝑚𝑞^{𝑚2(𝛼−𝛽)𝑚+}𝜋csc−𝛽𝜋Γ_𝑞−𝛽1−𝑞^1+𝛽×𝑞^{𝛽−𝛼−𝑚+1};𝑞_𝑚𝑞^𝛽+1;𝑞_𝑛𝑞;𝑞_𝑚𝑞;𝑞_𝑛×^𝑛_𝑗=0𝑞^−𝑛,𝑞^{𝛽−𝛼+1};𝑞_𝑗𝑞,𝑞^{𝛽−𝛼−𝑚+1};𝑞_𝑗𝑞^{𝑗(𝑛−𝑚)}, (36) 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 𝑞^{−𝑚𝛼−𝑛𝛽}𝑞^𝛼+1;𝑞_𝑚𝑞^𝛽+1;𝑞_𝑛𝑞;𝑞_𝑚𝑞;𝑞_𝑛×^𝑚𝑛_𝑘=0𝑗=0𝑞^−𝑚;𝑞_𝑘𝑞^−𝑛;𝑞_𝑗𝑞^𝛼+1,𝑞;𝑞_𝑘𝑞^𝛽+1,𝑞;𝑞_𝑗×^∞₀𝑥^{𝛽+𝑘+𝑗}𝑥;𝑞_∞𝑑𝑥=𝑞^{−𝑚𝛼−𝑛𝛽}𝑞^𝛼−𝛽;𝑞_𝑚𝑞^𝛽+1;𝑞_𝑛𝑞;𝑞_𝑚𝑞;𝑞_𝑛1−𝑞^1+𝛽×𝜋csc−𝛽𝜋Γ_𝑞𝑞−𝛽^(𝛽+1)𝑚𝑞²₂^𝛽+2𝜙₁𝑞^−𝑛,𝑞^{𝛽−𝛼+1}𝑞^{𝛽−𝛼−𝑚+1};𝑞, (37) 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. If 𝑓(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 𝑓(𝑥)=^𝑚_𝑛=0𝐶_𝑚𝑛ℒ_𝑛^(𝛼)𝑥;𝑞or𝑓(𝑥)=^𝑚_𝑛=0𝐷_𝑚𝑛ℳ_𝑛^(𝛼)𝑥;𝑞, (38) where 𝐶_𝑚𝑛 and 𝐷_𝑚𝑛 are the 𝑛th Fourier-Laguerre coefficients, and both of them are independent of 𝑥, then one has 𝐶_𝑚𝑛=Γ_𝑞(−𝛼)𝑞;𝑞_𝑛𝑞^𝑛Γ(𝛼+1)Γ(−𝛼)𝑞^𝛼+1;𝑞_𝑛1−𝑞^𝛼+1×^∞₀𝑥^𝛼−𝑥;𝑞_∞𝑓(𝑥)ℒ_𝑛^(𝛼)𝑥;𝑞𝑑𝑥,𝐷_𝑚𝑛=Γ_q(−𝛼)𝑞;𝑞_𝑛𝑞^{2𝛼𝑛−𝑛−𝛼+2}Γ(𝛼+1)Γ(−𝛼)𝑞^𝛼+1;𝑞_𝑛1−𝑞^𝛼+1×^∞₀𝑥^𝛼𝑥;𝑞_∞𝑓(𝑥)ℳ_𝑛^(𝛼)𝑥;𝑞𝑑𝑥. (39)(40)

Proof. Multiplying (38) by 𝑥^𝛼ℒ_𝑛(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 ^∞₀ℳ_𝑚^(𝛼)𝑥;𝑞ℳ_𝑛^(𝛼)𝑥;𝑞𝑥^𝛼𝑥;𝑞_∞𝑑𝑥=−𝑞^{2𝛼+2+𝑛−𝛼𝑛}𝑞^𝛼+1;𝑞_𝑛𝑞^−𝛼;𝑞_∞𝑞;𝑞_𝑛𝑞;𝑞_∞𝜋𝛿_𝑚,𝑛,sin𝜋𝛼 (41) so we also gain the proof of (40). The proof of Lemma 11 is complete.

Lemma 12. For min{𝛼,𝛽}>−1, one has ℒ_𝑛^(𝛼)𝑥𝑦;𝑞=^𝑛_𝑘=0𝑦𝑛+𝛼𝑛−𝑘^𝑘𝑞^{(𝛼−𝛽)𝑘}ℒ_𝑘^(𝛽)×𝑥;𝑞₂𝜙₁𝑞^𝑘−𝑛,𝑞^𝛽+1+𝑘𝑞^𝛼+𝑘+1;𝑦𝑞^{𝛼−𝛽+𝑛−𝑘},ℳ_𝑛^(𝛼)𝑥𝑦;𝑞=^𝑛_𝑘=0𝑦𝑛+𝛼𝑛−𝑘^𝑘𝑞^{(𝑘−𝑛+𝛽)𝑘−𝛼𝑛}ℳ_𝑘^(𝛽)×𝑥;𝑞₂𝜙₁𝑞^𝑘−𝑛,𝑞^𝛽+1+𝑘𝑞^𝛼+𝑘+1;𝑦𝑞. (42)(43)

Remark 13. Replacing 𝑥 by (1−𝑞)𝑥 and letting 𝑞→1, we have [15, equation (11)] 𝐿_𝑛^(𝛼)𝑥𝑦=^𝑛_𝑘=0𝑦𝑛+𝛼𝑛−𝑘^𝑘𝐿_𝑘^(𝛽)(𝑥)₂𝐹₁×.−𝑛+𝑘,𝛽+𝑘+1𝛼+𝑘+1;𝑦 (44) Setting 𝛼=𝛽, (42) and (43) reduce to (29) and (30), respectively.

Proof. Let Γ𝐴=^𝑞−𝛽𝑞;𝑞_𝑘𝑞^𝑘𝜋csc−𝛽𝜋𝑞^𝛽+1;𝑞_𝑘1−𝑞^1+𝛽,𝐵=1−𝑞^1+𝛽𝜋csc−𝛽𝜋Γ_𝑞.−𝛽 (45) By Lemmas 10 and 11, the coefficient of ℒ_𝑛^(𝛼)(𝑥𝑦;𝑞) expanded by ℒ_𝑛^(𝛽)(𝑥;𝑞)𝐶_𝑛𝑘=𝐴⋅^𝑛_𝑗=0𝑞^𝛼+1;𝑞_𝑛𝑦;𝑞_𝑛−𝑗𝑦^𝑗𝑞;𝑞_𝑛−𝑗𝑞^𝛼+1;𝑞_𝑗×^∞₀ℒ_𝑗^(𝛼)𝑥;𝑞ℒ_𝑘^(𝛽)𝑥𝑥;𝑞^𝛽−𝑥;𝑞_∞𝑑𝑥=𝐴𝐵⋅^𝑛_𝑗=0𝑞^𝛼+1;𝑞_𝑛𝑦;𝑞_𝑛−𝑗𝑦^𝑗𝑞;𝑞_𝑛−𝑗𝑞^𝛼+1;𝑞_𝑗×(−1)^𝑘+𝑗𝑞^{𝑗2+𝑘2(𝛼−𝛽)𝑗+}×𝑘𝑞𝛽−𝛼𝑗−𝑘𝑘+𝛽_−𝑘𝑗=𝐴𝐵⋅^𝑛−𝑘_𝑗=0𝑞^𝛼+1;𝑞_𝑛𝑦;𝑞_{𝑛−𝑗−𝑘}𝑦^𝑗+𝑘𝑞;𝑞_{𝑛−𝑗−𝑘}𝑞^𝛼+1;𝑞_𝑗+𝑘×𝑗𝑘()𝛽−𝛼𝑘+𝛽−1^𝑗𝑞^{𝑗2(𝛼−𝛽)(𝑗+𝑘)−𝑘+}𝑞=𝐴𝐵⋅^𝛼+1;𝑞_𝑛𝑦;𝑞_𝑛−𝑘𝑦^𝑘𝑞;𝑞_𝑛−𝑘𝑞^𝛼+1;𝑞_𝑘𝑞^{(𝛼−𝛽−1)𝑘}×𝑘𝑘+𝛽₃𝜙₂𝑞^𝑘−𝑛,𝑞^𝛼−𝛽𝑞,0_{𝑘−𝑛+1}𝑦,𝑞^𝛼+𝑘+1𝑞;𝑞=𝐴𝐵⋅^𝛼+1;𝑞_𝑛𝑦^𝑘𝑞;𝑞_𝑛−𝑘𝑞^𝛼+1;𝑞_𝑘𝑞^{(𝛼−𝛽−1)𝑘}×𝑘𝑘+𝛽₂𝜙₁𝑞^𝑘−𝑛,𝑞^𝛽+1+𝑘𝑞^𝛼+𝑘+1;𝑦𝑞^{𝛼−𝛽+𝑛−𝑘} (46) is equal to the right hand side of (42). Similarly, we have 𝐷_𝑛𝑘=𝐴⋅𝑞^{−2𝛽+2−2𝑘+𝛽𝑘}×^∞₀ℳ_𝑛^(𝛼)𝑥𝑦;𝑞ℳ_𝑘^(𝛽)𝑥;𝑞𝑥^𝛽𝑥;𝑞_∞𝑑𝑥=𝐴𝐵⋅𝑞^{−2𝛽+2−2𝑘+𝛽𝑘}×^𝑛_𝑗=0(−1)^𝑛−𝑗𝑞^𝛼+1;𝑞_𝑛1/𝑦;𝑞_𝑛−𝑗𝑦^𝑛𝑞;𝑞_𝑛−𝑗𝑞^𝛼+1;𝑞_𝑗𝑞^{𝑗2𝑛2+𝛼𝑗−−𝛼𝑛}×(−1)^𝑗+𝑘𝑞^{2+22𝑗+1𝑘+1−(𝛼+𝑗)𝑘+𝛽+2}×𝑘𝛽−𝛼𝑗−𝑘𝑘+𝛽=𝐴𝐵⋅𝑞^{𝑛2+2−2𝑘+𝛽𝑘−𝛼𝑛−𝑘+1−𝛼𝑘}×^𝑛−𝑘_𝑗=0(−1)^𝑛+𝑘𝑞^𝛼+1;𝑞_𝑛1/𝑦;𝑞_𝑛−𝑘𝑞^{𝑛−𝑗−𝑘+1};𝑞_𝑗𝑦^𝑛𝑞^{𝑛−𝑗−𝑘}/𝑦;𝑞_𝑗𝑞;𝑞_𝑛−𝑘𝑞^𝛼+1;𝑞_𝑗+𝑘×𝑞^{(𝑗+𝑘)2+𝛼(𝑗+𝑘)−(𝑗+𝑘)𝑘}×𝑞^{𝛽−𝛼−𝑗+1};𝑞_𝑗𝑞;𝑞_𝑗𝑘𝑘+𝛽=𝐴𝐵⋅𝑞^{𝑛2+2−2𝑘+𝛽𝑘−𝛼𝑛−𝑘+1}×(−1)^𝑛+𝑘𝑞^𝛼+1;𝑞_𝑛1/𝑦;𝑞_𝑛−𝑘𝑦^𝑛𝑞;𝑞_𝑛−𝑘𝑞^𝛼+1;𝑞_𝑘𝑘×𝑘+𝛽₂𝜙₂𝑞^𝑘−𝑛,𝑞^𝛼−𝛽𝑦𝑞^{𝑘+1−𝑛},𝑞^𝛼+𝑘+1;𝑦𝑞^𝛽+𝑘+2, (47) which is equivalent to the right hand side of (43) by (25) and simplification. The proof of Lemma 12 is complete.

Proof of Theorems 2 and 3. By using formula (42), the left hand side of (7) is equal to ^𝑚_𝑘=0𝑦𝑚+𝛼𝑚−𝑘^𝑘𝑞^{(𝛼−𝜇)𝑘}×₂𝜙₁𝑞^𝑘−𝑚,𝑞^𝜇+1+𝑘𝑞^𝛼+𝑘+1;𝑦𝑞^{𝛼−𝜇+𝑚−𝑘}×^𝑛_𝑗=0𝑧𝑛+𝛽𝑛−𝑗^𝑗𝑞^{(𝛽−𝜇)𝑗}×₂𝜙₁𝑞^𝑗−𝑛,𝑞^𝜇+1+𝑗𝑞^𝛽+𝑗+1;𝑧𝑞^{𝛽−𝜇+𝑛−𝑗}×^∞₀ℒ_𝑘^(𝜇)𝑥;𝑞ℒ_𝑗^(𝜇)𝑥𝑥;𝑞^𝜇−𝑥;𝑞_∞𝑑𝑥. (48) Similarly, with the help of formula (43), the left hand side of (8) equals ^𝑚_𝑘=0𝑦𝑚+𝛼𝑚−𝑘^𝑘𝑞₂^{(𝑘−𝑚+𝜇)𝑘−𝛼𝑚}𝜙₁×𝑞^𝑘−𝑚,𝑞^𝜇+1+𝑘𝑞^𝛼+𝑘+1×;𝑦𝑞_𝑛𝑗=0𝑧𝑛+𝛽𝑛−𝑗^𝑗𝑞^{(𝑗−𝑛+𝜇)𝑗−𝛽𝑛}×₂𝜙₁𝑞^𝑗−𝑛,𝑞^𝜇+1+𝑗𝑞^𝛽+𝑗+1×;𝑧𝑞_∞0ℳ_𝑘^(𝜇)𝑥;𝑞ℳ_𝑗^(𝜇)𝑥;𝑞𝑥^𝜇𝑥;𝑞_∞𝑑𝑥. (49) 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 of 𝑞-Hermite Polynomials

Doha [34, page 5460] deduced the following result by third-order recurrence relation of the coefficients.

Proposition 14 (see [34, equation (49)]). For 𝛼>−1 and 𝑛∈ℕ, one has 𝐻_𝑛(𝑥)=2^𝑛(1+𝛼)_𝑛×^𝑛_2𝑘=0𝐹₂−(𝑛−𝑘)2,−(𝑛−𝑘−1)2−(𝛼+𝑛)2,−(𝛼+𝑛−1)21;−4×(−𝑛)_𝑘𝐿_𝑘^(𝛼)(𝑥)(1+𝛼)_𝑘. (50)

In this section, we employ the technique of rearrangement of series ^∞∞_𝑛=0𝑘=0𝒜()=𝑘,𝑛_{∞𝑛𝑛=0𝑘=0}𝒜(𝑘,𝑛−𝑘)=^∞_𝑛=0^[𝑛/2]_𝑘=0𝒜(𝑘,𝑛−2𝑘) (51) to derive the following 𝑞-analogue of Proposition 14.

Theorem 15. For 𝛼>−1 and 𝑛∈ℕ, one has ℋ_𝑛𝑥;𝑞=𝑞^𝛼+1,𝑞;𝑞_𝑛1−𝑞^{𝑛−𝑛/2}_𝑘=0𝐶(𝑛,𝑘)ℒ_𝑘^(𝛼)𝑥;𝑞, (52) where 𝐶()=()𝑛,𝑘−1^𝑘𝑞^{−(𝑛2+𝑛+4𝛼𝑛+4𝑛𝑘−2𝑘2)/4}𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑛−𝑘×^{[(𝑛−𝑘)/2]}_𝑠=0(−1)^𝑠𝑞^𝑘−𝑛;𝑞_2𝑠1−𝑞^𝑠𝑞^{−(𝑛+𝛼)};𝑞_2𝑠−𝑞^−𝑛/2;𝑞^1/2_2𝑠𝑞;𝑞_𝑠, (53) and [𝑥] denotes the greatest integer not exceeding 𝑥.

Theorem 16. For 𝛼>−1 and 𝑛∈ℕ, one has 𝒢_𝑛𝑥;𝑞=𝑞^𝛼+1,𝑞;𝑞_𝑛1−𝑞^−𝑛/2𝑞^{−𝑛(𝑛−1)/4}×^𝑛_𝑘=0𝐷(𝑛,𝑘)ℳ_𝑘^(𝛼)𝑥;𝑞, (54) where 𝐷()=()𝑛,𝑘−1^𝑘𝑞^{𝑘2+𝛼𝑘}𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑛−𝑘×^{[(𝑛−𝑘)/2]}_𝑠=0(−1)^𝑠𝑞^𝑠21−𝑞^𝑠𝑞^𝑘−𝑛;𝑞_2𝑠𝑞^{(2𝑠2−𝑠−4𝑠𝑘−4𝑠𝛼)/2}𝑞;𝑞_𝑠𝑞^{−(𝑛+𝛼)};𝑞_2𝑠−𝑞^−𝑛/2;𝑞^1/2_2𝑠. (55)

Before the proof of Theorem 15 the following lemma is necessary.

Lemma 17. For 𝛼>−1 and 𝑛∈ℕ, one has 𝑥^𝑛=𝑞^𝛼+1,𝑞;𝑞_𝑛𝑞^{−(𝛼𝑛+𝑛2)𝑛}_𝑘=0(−1)^𝑘𝑞^2𝑛−𝑘ℒ_𝑘^(𝛼)𝑥;𝑞𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑛−𝑘,𝑥^𝑛=𝑞^𝛼+1,𝑞;𝑞_𝑛𝑞^𝑛𝑛_𝑘=0(−1)^𝑘𝑞^{𝑘2+𝛼𝑘}ℳ_𝑘^(𝛼)𝑥;𝑞𝑞;𝑞_𝑛−𝑘𝑞^𝛼+1;𝑞_𝑘. (56)

Proof. Letting 𝑓(𝑥)=𝑥^𝑚 in (38) and using the following fact [8, page 23]: ^∞₀ℒ_𝑛^(𝛼)𝑥𝑥;𝑞^𝛼+𝑚−𝑥;𝑞_∞=𝑞𝑑𝑥_−𝑚;𝑞_𝑛𝑞^𝛼+1;𝑞_𝑚Γ(−𝛼)Γ(𝛼+1)𝑞;𝑞_𝑛Γ_𝑞(−𝛼)𝑞^{2𝛼𝑚+𝑚+1}1−𝑞^−1−𝛼, (57) similarly, we deduce the explicit representation of (39) and (40), respectively, 𝐶_𝑚𝑛=𝑞^𝛼+1;𝑞_𝑚𝑞^−𝑚;𝑞_𝑛𝑞^𝑛𝑞^𝛼+1;𝑞_𝑛𝑞^{2𝛼𝑚+𝛼+1},𝐷_𝑚𝑛=𝑞^𝛼+1;𝑞_𝑚𝑞^−𝑚;𝑞_𝑛𝑞^𝑚(𝑛+1)𝑞^𝛼+1;𝑞_𝑛𝑞^−𝛼𝑛, (58) so we obtain the formula (56). The proof is complete.

Lemma 18 (see [7, equations (3.29.5) and (3.28.5)]). One has ^∞_𝑛=0𝑡^𝑛𝑞^1/2;𝑞^1/2_𝑛ℋ_𝑛𝑥;𝑞=−𝑥𝑡1−𝑞^−1/2;𝑞^1/2_∞−𝑡²;𝑞_∞∞𝑛=0,𝑡^𝑛𝑞^{𝑛(𝑛−1)/4}𝑞^1/2;𝑞^1/2_𝑛𝒢_𝑛=𝑡𝑥;𝑞²;𝑞_∞𝑥𝑡1−𝑞^−1/2;𝑞^1/2_∞. (59)

Proof. By using [7, equations (3.29.5) and (3.28.5)] ^∞∑_𝑛=0𝑞^𝑛2𝑞;𝑞_𝑛ℎ_𝑛𝑥;𝑞𝑡^𝑛=−𝑥𝑡;𝑞_∞−𝑡²;𝑞²_∞∞𝑛=0,ℎ_𝑛𝑥;𝑞𝑞;𝑞_𝑛𝑡^𝑛=𝑡²;𝑞²_∞𝑥𝑡;𝑞_∞. (60) and replacing, respectively, by ℋ_𝑛𝑥;𝑞=𝑞^{𝑛(𝑛−1)/4}ℎ_𝑛𝑥1−𝑞;𝑞^1/2,𝒢_𝑛𝑥;𝑞=𝑞^{−𝑛(𝑛−1)/4}ℎ_𝑛𝑥1−𝑞;𝑞^1/2, (61) we deduce the proof of Lemma 18. The proof is complete.

Proof of Theorem 15. From the generating function of ℋ_𝑛(𝑥;𝑞), we have ^∞_𝑛=0𝑡^𝑛𝑞^1/2;𝑞^1/2_𝑛ℋ_𝑛=𝑥;𝑞_∞𝑛=0𝑞^{𝑛(𝑛−1)/4}(𝑥𝑡)^𝑛𝑞^1/2;𝑞^1/2_𝑛1−𝑞^∞−𝑛/2_𝑠=0−𝑡^2𝑠𝑞;𝑞_𝑠=^∞_𝑛,𝑠=0𝑞^{𝑛(𝑛−1)/4}𝑡^𝑛+2𝑠(−1)^𝑠𝑥^𝑛𝑞^1/2;𝑞^1/2_𝑛𝑞;𝑞_𝑠=^∞_𝑛,𝑠=0𝑞^{𝑛(𝑛−1)/4}𝑡^𝑛+2𝑠(−1)^𝑠1−𝑞^−𝑛/2𝑞^1/2;𝑞^1/2_𝑛𝑞;𝑞_𝑠×𝑞^𝛼+1,𝑞;𝑞_𝑛𝑞^{−(𝛼𝑛+𝑛2)}×^𝑛_𝑘=0(−1)^𝑗𝑞^2𝑛−𝑘ℒ_𝑘^(𝛼)𝑥;𝑞𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑛−𝑘=^∞_{𝑛,𝑠,𝑘=0}𝑞^{(𝑛+𝑘)(𝑛+𝑘−1)/4}(−1)^𝑠𝑡^{𝑛+2𝑠+𝑘}𝑞^𝛼+1;𝑞_𝑛+𝑘×𝑞^{−[𝛼(𝑛+𝑘)+(𝑛+𝑘)2]}1−𝑞^{−(𝑛+𝑘)/2}×𝑞^1/2;𝑞^1/2_𝑛+𝑘𝑞;𝑞_𝑠⁻¹×(−1)^𝑘𝑞^𝑛2ℒ_𝑘^(𝛼)𝑥;𝑞𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑛=^∞_𝑛,𝑘=0^[𝑛/2]_𝑠=0𝑞^{(𝑛+𝑘−2𝑠)(𝑛+𝑘−2𝑠−1)/4}(−1)^𝑠𝑡^𝑛+𝑘×𝑞^𝛼+1;𝑞_{𝑛−2𝑠+𝑘}𝑞^{−[𝛼(𝑛−2𝑠+𝑘)+(𝑛−2𝑠+𝑘)2]}×𝑞^1/2;𝑞^1/2_{𝑛−2𝑠+𝑘}𝑞;𝑞_𝑠⁻¹×1−𝑞^{−(𝑛+𝑘−2𝑠)/2}(−1)^𝑘𝑞^{2𝑛−2𝑠}ℒ_𝑘^(𝛼)𝑥;𝑞𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑛−2𝑠=^∞_𝑛,𝑘=0(−1)^𝑘𝑡^𝑛+𝑘ℒ_𝑘^(𝛼)𝑥;𝑞1−𝑞^{−(𝑛+𝑘)/2}𝑞^𝛼+1;𝑞_𝑘×^[𝑛/2]_𝑠=0(−1)^𝑠𝑞^𝛼+1,𝑞;𝑞_{𝑛−2𝑠+𝑘}1−𝑞^𝑠𝑞^1/2;𝑞^1/2_{𝑛+𝑘−2𝑠}𝑞;𝑞_𝑛−2𝑠𝑞;𝑞_𝑠×𝑞^{(𝑛+𝑘−2𝑠)(𝑛+𝑘−2𝑠−1)/4}×𝑞^{−[𝛼(𝑛+𝑘−2𝑠)+(𝑛+𝑘−2𝑠)22]+𝑛−2𝑠}=^∞_𝑛,𝑘=0(−1)^𝑘𝑡^𝑛+𝑘ℒ_𝑘^(𝛼)𝑥;𝑞1−𝑞^{−(𝑛+𝑘)/2}×𝑞^𝛼+1;𝑞_𝑛+𝑘−𝑞^1/2;𝑞^1/2_𝑛+𝑘×𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑛⁻¹×^[𝑛/2]_𝑠=0(−1)^𝑠𝑞^−𝑛;𝑞_2𝑠1−𝑞^𝑠𝑞^{−(𝑛+𝑘+𝛼)};𝑞_2𝑠−𝑞^{−(𝑛+𝑘)/2};𝑞^1/2_2𝑠𝑞;𝑞_𝑠×𝑞^{(𝑛+𝑘−2𝑠)(𝑛+𝑘−2𝑠−1)/4−[𝛼(𝑛+𝑘−2𝑠)+(𝑛+𝑘−2𝑠)2]}×𝑞^{+2𝑛−2𝑠−2𝑠(𝛼+𝑘)−𝑠(𝑛+𝑘)+𝑠(2𝑠−1)/2}=^∞_𝑛=0𝑞^𝛼+1;𝑞_𝑛−𝑞^1/2;𝑞^1/2_𝑛1−𝑞^−𝑛/2𝑡^𝑛×^𝑛_𝑘=0(−1)^𝑘ℒ_𝑘^(𝛼)𝑥;𝑞𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑛−𝑘×^{[(𝑛−𝑘)/2]}_𝑠=0(−1)^𝑠𝑞^𝑘−𝑛;𝑞_2𝑠1−𝑞^𝑠𝑞^{−(𝑛+𝛼)};𝑞_2𝑠−𝑞^−𝑛/2;𝑞^1/2_2𝑠𝑞;𝑞_𝑠×𝑞^{(𝑛−2𝑠)(𝑛−2𝑠−1)/4−[𝛼(𝑛−2𝑠)+(𝑛−2𝑠)2]}×𝑞^{+2𝑛−𝑘−2𝑠−2𝑠(𝛼+𝑘)−𝑠𝑛+𝑠(2𝑠−1)/2}. (62) Comparing the coefficients of 𝑡^𝑛 on both sides of (62), we obtain the results.

Proof of Theorem 16. From the generating function of 𝒢_𝑛(𝑥;𝑞), we have ^∞_𝑛=0𝑡^𝑛𝑞^{𝑛(𝑛−1)/4}𝑞^1/2;𝑞^1/2_𝑛𝒢_𝑛=𝑥;𝑞_{∞𝑛,𝑠=0}(−1)^𝑠𝑞^𝑠2𝑥^𝑛1−𝑞^−𝑛/2𝑡^𝑛+2𝑠𝑞^1/2;𝑞^1/2_𝑛𝑞;𝑞_𝑠=^∞_{𝑛,𝑠,𝑘=0}(−1)^𝑠𝑞^𝑠2+𝑛1−𝑞^{−(𝑛+𝑘)/2}𝑡^{𝑛+𝑘+2𝑠}×𝑞^𝛼+1;𝑞_𝑛+𝑘(−1)^𝑘𝑞^{𝑘2+𝛼𝑘}ℳ_𝑘^(𝛼)×𝑞𝑥;𝑞_1/2;𝑞^1/2_𝑛+𝑘𝑞;𝑞_𝑛𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑠⁻¹=^∞_𝑛,𝑘=0(−1)^𝑘𝑞^{𝑘2+𝛼𝑘+𝑛}ℳ_𝑘^(𝛼)𝑥;𝑞1−𝑞^{−(𝑛+𝑘)/2}𝑡^𝑛+𝑘𝑞^𝛼+1;𝑞_𝑘×^[𝑛/2]_𝑠=0(−1)^𝑠𝑞^{𝑠2−2𝑠}1−𝑞^𝑠𝑞^𝛼+1;𝑞_{𝑛−2𝑠+𝑘}𝑞^1/2;𝑞^1/2_{𝑛+𝑘−2𝑠}𝑞;𝑞_𝑛−2𝑠𝑞;𝑞_𝑠=^∞_𝑛,𝑘=0(−1)^𝑘𝑞^{𝑘2+𝛼𝑘+𝑛}ℳ_𝑘^(𝛼)𝑥;𝑞1−𝑞^{−(𝑛+𝑘)/2}𝑡^𝑛+𝑘𝑞^𝛼+1;𝑞_𝑘×𝑞^𝛼+1;𝑞_𝑛+𝑘−𝑞^1/2;𝑞^1/2_𝑛+𝑘𝑞;𝑞_𝑛×^[𝑛/2]_𝑠=0(−1)^𝑠𝑞^{𝑠2−2𝑠}1−𝑞^𝑠𝑞^−𝑛;𝑞_2𝑠𝑞;𝑞_𝑠𝑞^{−(𝑛+𝑘+𝛼)};𝑞_2𝑠−𝑞^{−(𝑛+𝑘)/2};𝑞^1/2_2𝑠×𝑞^{−2𝑠(𝑘+𝛼)+𝑠(2𝑠−1)/2}=^∞_𝑛=0𝑞^𝛼+1;𝑞_𝑛−𝑞^1/2;𝑞^1/2_𝑛1−𝑞^−𝑛/2𝑡^𝑛×^𝑛_𝑘=0(−1)^𝑘𝑞^{𝑘2+𝛼𝑘+𝑛−𝑘}ℳ_𝑘^(𝛼)𝑥;𝑞𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑛−𝑘×^{[(𝑛−𝑘)/2]}_𝑠=0(−1)^𝑠𝑞^{𝑠2−2𝑠}1−𝑞^𝑠𝑞^𝑘−𝑛;𝑞_2𝑠𝑞;𝑞_𝑠𝑞^{−(𝑛+𝛼)};𝑞_2𝑠−𝑞^−𝑛/2;𝑞^1/2_2𝑠×𝑞^{−2𝑠(𝑘+𝛼)+𝑠(2𝑠−1)/2}. (63) Equating the coefficients of 𝑡^𝑛 on both sides of (63), we obtain the results.

Proof of Corollary 23. In view of the fact that 𝑞^𝛽;𝑞_2𝑛=𝑞^𝛽/2,−𝑞^𝛽/2,𝑞^(𝛽+1)/2,−𝑞^(𝛽+1)/2;𝑞_𝑛,𝛽∈ℕ, (64) we have lim_𝑞→1^{[(𝑛−𝑘)/2]}_𝑠=0(−1)^𝑠𝑞^𝑘−𝑛;𝑞_2𝑠1−𝑞^𝑠𝑞^{𝑠2−2𝑠}𝑞^{−(𝑛+𝛼)};𝑞_2𝑠−𝑞^−𝑛/2;𝑞^1/2_2𝑠𝑞;𝑞_𝑠=lim_𝑞→1^{[(𝑛−𝑘)/2]}_𝑠=0(−1)^𝑠𝑞^{−(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;𝑞^1/2_2𝑠𝑞;𝑞_𝑠=₂𝐹₂−(𝑛−𝑘)2,−(𝑛−𝑘−1)2−(𝛼+𝑛)2,−(𝛼+𝑛−1)21;−4,lim_{𝑥→1−𝑞𝑥,𝑞→1}𝑞^𝛼+1;𝑞_𝑛𝑞;𝑞_𝑛1−𝑞^𝑛𝑛_𝑘=0(−1)^𝑘ℒ_𝑘^(𝛼)𝑥;𝑞𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑛−𝑘×𝑞^{−𝑛2/4−𝑛/4−𝛼𝑛−𝑛𝑘+𝑘2/2}=lim_{𝑥→1−𝑞𝑥,𝑞→1}𝑞^𝛼+1,𝑞;𝑞_𝑛1−𝑞^𝑛𝑞^{−𝑛(𝑛−1)/4}×^𝑛_𝑘=0(−1)^𝑘𝑞^{𝑘2+𝛼𝑘}ℳ_𝑘^(𝛼)𝑥;𝑞𝑞^𝛼+1;𝑞_𝑘𝑞;𝑞_𝑛−𝑘=(1+𝛼)_𝑛^𝑛_𝑘=0(−𝑛)_𝑘𝐿_𝑘^(𝛼)(𝑥)(1+𝛼)_𝑘. (65) Combing (65) and (2), we deduce (50). The proof of Corollary 23 is complete.

4. Integrals Related to Generalized 𝑞-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 the 𝑞-polynomials {ℋ_𝑛^(𝜇)(𝑥;𝑞)}, which are defined by the relations (16), satisfies the orthogonality relation ^∞_−∞ℋ_𝑚^(𝜇)𝑥;𝑞ℋ_𝑛^(𝜇)|𝑥|𝑥;𝑞^2𝜇−𝑥²;𝑞_∞=𝜋𝑑𝑥𝑞cos𝜋𝜇^1/2−𝜇;𝑞_∞𝑞;𝑞_∞𝑞^{−𝑛/2−𝜇𝜃𝑛}×𝑞;𝑞_[𝑛/2]𝑞^𝜇+1/2;𝑞_[(𝑛+1)/2]𝛿_𝑚𝑛 (66) on the whole real line ℝ, where 𝜃_𝑛=𝑛−2[𝑛/2].

In this section, we will further consider multivariate 𝑞-Hermite polynomials by Theorems 2 and 3.

Theorem 20. For 𝜃_𝑛=𝑛−2[𝑛/2], one has ^∞_−∞ℋ_𝑚^(𝛼)𝑥𝑦;𝑞ℋ_𝑛^(𝛽)|𝑥|𝑥𝑧;𝑞^2𝜇−𝑥²;𝑞_∞=()𝑑𝑥−1^{[𝑚/2]+[𝑛/2]}𝑞;𝑞_[𝑚/2]𝑞;𝑞_[𝑛/2]×1−𝑞^{𝜇+1/2+𝜃𝑛}𝜋csc−𝜇+1/2−𝜃_𝑛𝜋Γ_𝑞−𝜇+1/2−𝜃_𝑛×^{min{[𝑚/2],[𝑛/2]}}_𝑘=0𝑞^{(𝛼+𝛽−2𝜇−1)𝑘}𝑦𝑧^{2𝑘+𝜃𝑛}×₂𝜙₁𝑞^{𝑘−[𝑚/2]},𝑞^{𝜇+1/2+𝑘+𝜃𝑛}𝑞^{𝛼+𝑘+1/2+𝜃𝑛};𝑦²𝑞^{𝛼−𝜇+[𝑚/2]−𝑘}×₂𝜙₁𝑞^{𝑘−[𝑛/2]},𝑞^{𝜇+1/2+𝑘+𝜃𝑛}𝑞^{𝛽+𝑘+1/2+𝜃𝑛};𝑧²𝑞^{𝛽−𝜇+[𝑛/2]−𝑘}×𝑚21+𝛼−2+𝜃_𝑛𝑚2𝑛−𝑘21+𝛽−2+𝜃_𝑛𝑛2×1−𝑘𝑘+𝜇−2+𝜃_𝑛𝑘. (67)

Theorem 21. For 𝜃_𝑛=𝑛−2[𝑛/2], one has ^∞_−∞𝒢_𝑚^(𝛼)𝑥𝑦;𝑞𝒢_𝑛^(𝛽)𝑥𝑧;𝑞|𝑥|^2𝜇𝑥²;𝑞_∞=𝑑𝑥𝑞;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−𝜃_𝑛𝜋Γ_𝑞−𝜇+1/2−𝜃_𝑛×^{min{[𝑚/2],[𝑛/2]}}_𝑘=0𝑞^{(2𝑘+𝜇+1/2−[𝑚/2]−[𝑛/2]+𝜃𝑛)𝑘}×𝑦𝑧^{2𝑘+𝜃𝑛}₂𝜙₁𝑞^{𝑘−[𝑚/2]},𝑞^{𝜇+1/2+𝑘+𝜃𝑛}𝑞^{𝛼+𝑘+1/2+𝜃𝑛};𝑦²𝑞×₂𝜙₁𝑞^{𝑘−[𝑛/2]},𝑞^{𝜇+1/2+𝑘+𝜃𝑛}𝑞^{𝛽+𝑘+1/2+𝜃𝑛};𝑧²𝑞×𝑚21+𝛼−2+𝜃_𝑛𝑚2𝑛−𝑘21+𝛽−2+𝜃_𝑛𝑛2×1−𝑘𝑘+𝜇−2+𝜃_𝑛𝑘. (68)

Remark 22. For 𝛼=𝛽=𝜇 and 𝑦=𝑧=1, Theorems 20 and 21 reduce to Proposition 19 and Corollary 23 respectively.

Corollary 23. The sequence of the 𝑞-polynomials {𝒢_𝑛^(𝜇)(𝑥;𝑞)}, which are defined by the relations (16), satisfies the orthogonality relation ^∞_−∞𝒢_𝑚^(𝜇)𝑥;𝑞𝒢_𝑛^(𝜇)|𝑥|𝑥;𝑞^2𝜇−𝑥²;𝑞_∞=𝜋𝑑𝑥𝑞cos𝜋𝜇^1/2−𝜇;𝑞_∞𝑞;𝑞_∞×𝑞^{2𝜇+3/2−[𝑛/2](𝜇−3/2+𝜃𝑛)+𝜃𝑛}×𝑞;𝑞_[𝑛/2]𝑞^𝜇+1/2;𝑞_[(𝑛+1)/2]𝛿_𝑚𝑛 (69) on the whole real line ℝ, where 𝜃_𝑛=𝑛−2[𝑛/2].

Proof of Theorems 20 and 21. Let ℐ(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 ^∞_−∞ℋ^(𝛼)_2𝑚𝑥𝑦;𝑞ℋ^(𝛽)_2𝑛|𝑥|𝑥𝑧;𝑞^2𝜇−𝑥²;𝑞_∞=()𝑑𝑥−1^𝑚+𝑛𝑞;𝑞_𝑚𝑞;𝑞_𝑛×^∞_−∞ℒ_𝑚^(𝛼−1/2)𝑥²𝑦²;𝑞ℒ_𝑛^(𝛽−1/2)×𝑥²𝑧²|𝑥|;𝑞^2𝜇−𝑥²;𝑞_∞()𝑑𝑥=2−1^𝑚+𝑛𝑞;𝑞_𝑚𝑞;𝑞_𝑛×^∞₀ℒ_𝑚^(𝛼−1/2)𝑥²𝑦²;𝑞ℒ_𝑛^(𝛽−1/2)×𝑥²𝑧²|𝑥|;𝑞^2𝜇−𝑥²;𝑞_∞=()𝑑𝑥−1^𝑚+𝑛𝑞;𝑞_𝑚𝑞;𝑞_𝑛×^∞₀ℒ_𝑚^(𝛼−1/2)𝑦²𝑡;𝑞ℒ_𝑛^(𝛽−1/2)𝑧²𝑡𝑡;𝑞^𝜇−1/2−𝑡;𝑞_∞=()𝑑𝑡−1^𝑚+𝑛𝑞;𝑞_𝑚𝑞;𝑞_𝑛1×ℐ𝛼−2,𝑦²1,𝑚;𝛽−2,𝑧²1,𝑛;𝜇−2_∞−∞,ℋ^(𝛼)_2𝑚+1𝑥𝑦;𝑞ℋ^(𝛽)_2𝑛+1|𝑥|𝑥𝑧;𝑞^2𝜇−𝑥²;𝑞_∞()𝑑𝑥=2−1^𝑚+𝑛𝑦𝑧𝑞;𝑞_𝑚𝑞;𝑞_𝑛×^∞₀ℒ_𝑚^(𝛼+1/2)𝑥²𝑦²;𝑞ℒ_𝑛^(𝛽+1/2)𝑥²𝑧²×|𝑥|;𝑞_2𝜇+2−𝑥²;𝑞_∞=()𝑑𝑥−1^𝑚+𝑛𝑦𝑧𝑞;𝑞_𝑚𝑞;𝑞_𝑛×^∞₀ℒ_𝑚^(𝛼+1/2)𝑦²𝑡;𝑞ℒ_𝑛^(𝛽+1/2)𝑧²𝑡𝑡;𝑞^𝜇+1/2−𝑡;𝑞_∞=()𝑑𝑡−1^𝑚+𝑛𝑦𝑧𝑞;𝑞_𝑚𝑞;𝑞_𝑛1×ℐ𝛼+2,𝑦²1,𝑚;𝛽+2,𝑧²1,𝑛;𝜇+2. (70)(71) 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 ^∞_−∞𝒢^(𝛼)_2𝑚𝑥𝑦;𝑞𝒢^(𝛽)_2𝑛𝑥𝑧;𝑞|𝑥|^2𝜇−𝑥²;𝑞_∞𝑑𝑥=𝑞^{−2−2𝑚+1𝑛+1}𝑞;𝑞_𝑚𝑞;𝑞_𝑛𝒥×1𝛼−2,𝑦²1,𝑚;𝛽−2,𝑧²1,𝑛;𝜇−2_∞−∞,𝒢^(𝛼)_2𝑚+1𝑥𝑦;𝑞𝒢^(𝛽)_2𝑛+1𝑥𝑧;𝑞|𝑥|^2𝜇−𝑥²;𝑞_∞𝑑𝑥=𝑞^{−2−2𝑚+1𝑛+1}𝑦𝑧𝑞;𝑞_𝑚𝑞;𝑞_𝑛1×𝒥𝛼+2,𝑦²1,𝑚;𝛽+2,𝑧²1,𝑛;𝜇+2, (72) which are two cases of Theorem 21; thus we obtain the proof.

Proof of Proposition 19 and Corollary 23. Let us consider first that the case of both 𝑚 and 𝑛 is even, and just take 𝛼=𝛽=𝜇 and 𝑦=𝑧=1 in Theorem 20. We have ^∞_−∞ℋ^(𝜇)_2𝑚𝑥;𝑞ℋ^(𝜇)_2𝑛|𝑥|𝑥;𝑞^2𝜇−𝑥²;𝑞_∞=()𝑑𝑥−1^𝑚+𝑛𝑞;𝑞_𝑚𝑞;𝑞_𝑛1−𝑞^𝜇+1/2×𝜋csc−𝜇+1/2𝜋Γ_𝑞−𝜇+1/2^{min{𝑚,𝑛}}_𝑘=0𝑞^−𝑘×₁𝜙₀𝑞^𝑘−𝑚−;𝑞^𝑚−𝑘₁𝜙₀𝑞^𝑘−𝑛−;𝑞^𝑛−𝑘×1𝑚+𝛼−21𝑚−𝑘𝑛+𝛽−21𝑛−𝑘𝑘+𝜇−2𝑘=𝑞;𝑞_𝑛𝑞;𝑞_𝑛1−𝑞^𝜇+1/2×𝜋csc−𝜇+1/2𝜋Γ_𝑞𝑞−𝜇+1/2^−𝑛1𝑛+𝜇−2𝑛𝛿_𝑚𝑛=𝜋𝑞cos𝜇𝜋^1/2−𝜇;𝑞_∞𝑞;𝑞_∞𝑞^−𝑛𝑞;𝑞_𝑛𝑞^𝜇+1/2;𝑞_𝑛𝛿_𝑚𝑛,_∞−∞ℋ^(𝜇)_2𝑚+1𝑥;𝑞ℋ^(𝜇)_2𝑛+1|𝑥|𝑥;𝑞^2𝜇−𝑥²;𝑞_∞=𝑑𝑥𝑞;𝑞_𝑛𝑞;𝑞_𝑛1−𝑞^𝜇+3/2×𝜋csc−𝜇−1/2𝜋Γ_𝑞𝑞−𝜇−1/2^−𝑛1𝑛+𝜇+2𝑛𝛿_𝑚𝑛𝜋=−𝑞cos𝜇𝜋^{−𝜇−1/2};𝑞_∞𝑞;𝑞_∞×𝑞^−𝑛𝑞;𝑞_𝑛𝑞;𝑞_{𝑛+𝜇+1/2}𝑞;𝑞_𝜇+1/2𝛿_𝑚𝑛=𝜋𝑞cos𝜇𝜋^1/2−𝜇;𝑞_∞𝑞;𝑞_∞×𝑞^{−𝑛−𝜇−1/2}𝑞;𝑞_𝑛𝑞^𝜇+1/2;𝑞_𝑛+1𝛿_𝑚𝑛. (73)(74) 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.

Acknowledgments

The author thanks Professor Roelof Koekoek, who provided the hardcopy of his paper kindly. This work was supported by Tianyuan Special Funds of the National Natural Science Foundation of China (no. 11226298), China Postdoctoral Science Foundation (no. 2012M521155), Zhejiang Projects for Postdoctoral Research Preferred Funds (no. Bsh1201021), and Zhejiang Provincial Natural Science Foundation of China (no. LQ13A010021).