1 Introduction

The objective of this paper is to extend the work of Liu [14, 15] and Liu and Zeng [17]. These authors have found a q-difference equation related to Rogers–Szegö polynomials [21] which can be used to find interesting transformation formulas. We do the same analysis for the more general Al-Salam–Carlitz polynomials [8]. We apply this approach to provide a generating function for Al-Salam–Carlitz polynomials, generalize Ramanujan’s q-beta integrals and the q-Chu–Vandermonde summation formula. For further information about basic hypergeometric series and q-orthogonal polynomials, see [2, 11, 12, 23].

In this paper, we follow the notations and terminology in [9] and suppose that \(0<q<1\). The basic hypergeometric series \({}_r\phi _s\)

$$\begin{aligned} {}_r\phi _s\biggl [\begin{array}{l} \begin{array}{ccc} a_1,a_2,\ldots ,a_r\\ b_1,b_2,\ldots ,b_s \end{array} \end{array};q,z\biggr ]=\sum _{n=0}^\infty \frac{\bigl (a_1,a_2,\ldots ,a_r;q\bigr )_n}{\bigl (q,b_1,b_2,\ldots ,b_s;q\bigr )_n} \Bigl [(-1)^nq^{n\atopwithdelims ()2}\Bigr ]^{1+s-r}z^n \end{aligned}$$
(1.1)

converges absolutely for all z if \(r\le s\) and for \(\left|z\right|<1\) if \(r=s+1\) and for terminating. The compact factorials of \({}_r\phi _s\) are defined, respectively, by

$$\begin{aligned} (a;q)_0=1,\quad (a;q)_n=\prod _{k=0}^{n-1}(1-aq^k),\quad (a;q)_\infty =\prod _{k=0}^\infty (1-aq^k) \end{aligned}$$
(1.2)

and \((a_1,a_2,\cdots ,a_m;q)_n=(a_1;q)_n(a_2;q)_n\cdots (a_m;q)_n\), where \(m\in \mathbb N:=\{1,2,3,\ldots \}\,\text {and}\, n\in \mathbb N_0:=\mathbb N\cup \{0\}\).

The Rogers–Szegö polynomials were introduced by Szegö in 1926 but were already studied earlier by Rogers in 1894–1895. A good definition can be found in the book by Barry Simon [20, Ex. (1.6.5), pp. 77–87].

The homogeneous Rogers–Szegö polynomials [18, p. 3]

$$\begin{aligned} h_n(b,c\arrowvert q)=\sum _{k=0}^n\left[ \begin{array}{l} n \\ k \\ \end{array}\right] b^kc^{n-k}\quad \text {and}\quad g_n(b,c\arrowvert q)=\sum _{k=0}^n\left[ \begin{array}{l} n \\ k \\ \end{array}\right] q^{k(k-n)}b^kc^{n-k}. \end{aligned}$$
(1.3)

The Al-Salam–Carlitz polynomials were introduced by Al-Salam and Carlitz in 1965 [1, Eqs. (1.11) and (1.15)]

$$\begin{aligned} \phi _n^{(a)}(x\arrowvert q)=\sum _{k=0}^n\left[ \begin{array}{l} n \\ k \\ \end{array}\right] (a;q)_kx^k \quad \text {and}\quad \psi _n^{(a)}(x\arrowvert q)=\sum _{k=0}^n\left[ \begin{array}{l} n \\ k \\ \end{array}\right] q^{k(k-n)}x^k(aq^{1-k};q)_k. \end{aligned}$$
(1.4)

They play important roles in the theory of q-orthogonal polynomials. In fact, there are two families of these polynomials: one with continuous orthogonality and another with discrete orthogonality. They are given explicitly in the book of Koekoek–Swarttouw–Lesky [13, Eqs. (14.24) and (14.25), pp. 534–540].

The generalized Al-Salam–Carlitz polynomials [7, Eq. (4.7)]

$$\begin{aligned} \phi _n^{(a,b,c)}(x,y\arrowvert q)=\sum _{k=0}^n\left[ \begin{array}{l} n \\ k \\ \end{array}\right] \frac{(a,b;q)_k}{(c;q)_k}x^ky^{n-k}\quad \text {and}\nonumber \\ \psi _n^{(a,b,c)}(x,y\arrowvert q)=\sum _{k=0}^n\left[ \begin{array}{l} n \\ k \\ \end{array}\right] \frac{(a,b;q)_k}{(c;q)_k}(-1)^kq^{{{k+1}\atopwithdelims ()2}-nk}x^ky^{n-k}, \end{aligned}$$
(1.5)

whose generating functions are [7, Eqs. (4.10) and (4.11)]

(1.6)
(1.7)

Liu [14, 15] obtained several important results by using the following q-difference equations. Liu and Zeng [17] provide further applications of these q-difference methods to q-orthogonal polynomials.

Proposition 1

([17, Eqs. (1.7) and (1.8)]) Let f(a, b) be a two-variable analytic function at \((0,0)\in \mathbb C^2\). Then

  1. (A)

    f can be expanded in terms of \(h_n(a,b\arrowvert q)\) if and only if f satisfies the functional equation

    $$\begin{aligned} bf(aq,b)-af(a,bq)=(b-a)f(a,b). \end{aligned}$$
    (1.8)
  2. (B)

    f can be expanded in terms of \(g_n(a,b\arrowvert q)\) if and only if f satisfies the functional equation

    $$\begin{aligned} af(aq,b)-bf(a,bq)=(a-b)f(aq,bq). \end{aligned}$$
    (1.9)

The method of q-difference equation is an effective way to obtain many results in q-series. For more information, please refer to [6, 7, 14, 15].

Theorem 2

Let f(a, b, c, x, y) be a five-variable analytic function in a neighbourhood of   \((a,b,c,x,y)=(0,0,0,0,0)\in \mathbb C^5\).

  1. (I)

    If f(a, b, c, x, y) can be expanded in terms of \(\phi _n^{(a,b,c)}(x,y\arrowvert q)\) if and only if

    $$\begin{aligned}&y\left[ f(a,b,c,x,y)-\Bigl (1+q^{-1}c\Bigr )f(a,b,c,qx,y)+q^{-1}cf\bigl (a,b,c,q^2x,y\bigr )\right] \nonumber \\&\quad =x\Biggl \{\Bigl [f(a,b,c,x,y)-f(a,b,c,x,qy)\Bigr ]\nonumber \\ {}&\qquad \qquad -(a+b)\Bigl [f(a,b,c,qx,y)-f(a,b,c,qx,qy)\Bigr ]\nonumber \\&\qquad \qquad +ab\Bigl [f\bigl (a,b,c,q^2x,y\bigr )-f\bigl (a,b,c,q^2x,qy\bigr )\Bigr ]\Biggr \}. \end{aligned}$$
    (1.10)
  2. (II)

    If f(a, b, c, x, y) can be expanded in terms of \(\psi _n^{(a,b,c)}(x,y\arrowvert q)\) if and only if

    $$\begin{aligned}&q^{-1}y\left[ f(a,b,c,x,y)-\bigl (1+q^{-1}c\bigr )f\bigl (a,b,c,qx,y\bigr )+q^{-1}cf\bigl (a,b,c,q^2x,y\bigr )\right] \nonumber \\&\quad =x\Biggl \{\Bigl [f(a,b,c,x,y)-f\bigl (a,b,c,x,q^{-1}y\bigr )\Bigr ] \nonumber \\ {}&\qquad \qquad -(a+b)\Bigl [f(a,b,c,qx,y)-f\bigl (a,b,c,qx,q^{-1}y\bigr )\Bigr ]\nonumber \\&\qquad \qquad +ab\Bigl [f\bigl (a,b,c,q^2x,y\bigr )-f\bigl (a,b,c,q^2x,q^{-1}y\bigr )\Bigr ]\Biggr \}. \end{aligned}$$
    (1.11)

Remark 3

For \(a=b=c=0\) in Theorem 2, Eqs. (1.10) and (1.11) reduce to (1.8) and (1.9), respectively.

To determine if a given function is an analytic function in several complex variables, we often use the following Hartogs’s theorem. For more information, please refer to Taylor [22, p. 28] and Liu [16, Theorem 1.8].

Proposition 4

(Hartogs’s theorem [10, p. 15]) If a complex-valued function is holomorphic (analytic) in each variable separately in an open domain \(D\subseteq \mathbb C^n\), then it is holomorphic (analytic) in D.

In order to prove Theorem 2, we need the following fundamental property of several complex variables.

Proposition 5

([19, p. 5, Proposition 1]) If \(f(x_1,x_2,\ldots ,x_k)\) is analytic at the origin \((0,0,\ldots ,0)\in \mathbb C^k\), then, f can be expanded in an absolutely convergent power series,

$$\begin{aligned} f(x_1,x_2,\ldots ,x_k)=\sum _{n_1,n_2,\ldots ,n_k=0}^\infty \alpha _{n_1,n_2,\ldots ,n_k}x_1^{n_1}x_2^{n_2}\cdots x_k^{n_k}. \end{aligned}$$

Proof of Theorem 2

From the Hartogs’s theorem and the theory of several complex variables (see Propositions 4 and 5), we assume that

$$\begin{aligned} f(a,b,c,x,y)=\sum _{k=0}^\infty A_k(a,b,c,y)x^k. \end{aligned}$$
(1.12)

On one hand, substituting Eq. (1.12) into (1.10) yields

$$\begin{aligned}&y\Biggl [\sum _{k=0}^\infty A_k(a,b,c,y)x^k-\Bigl (1+q^{-1}c\Bigr )\sum _{k=0}^\infty A_k(a,b,c,y)(qx)^k\nonumber \\&\qquad +q^{-1}c\sum _{k=0}^\infty A_k(a,b,c,y)\bigl (q^2x\bigr )^k\Biggr ]\nonumber \\&\qquad \quad =x\Biggl \{\Biggl [\sum _{k=0}^\infty A_k(a,b,c,y)x^k-\sum _{k=0}^\infty A_k(a,b,c,qy)x^k\Biggr ]\nonumber \\&\qquad -(a+b)\Biggl [\sum _{k=0}^\infty A_k(a,b,c,y)(qx)^k-\sum _{k=0}^\infty A_k(a,b,c,qy)(qx)^k\Biggr ]\nonumber \\&\qquad +ab\Biggl [\sum _{k=0}^\infty A_k(a,b,c,y)\bigl (q^2x\bigr )^k-\sum _{k=0}^\infty A_k(a,b,c,qy)\bigl (q^2x\bigr )^k\Biggr ]\Biggr \}. \end{aligned}$$
(1.13)

By equating coefficients of \(x^k\) on both sides of Eq. (1.13), we have

$$\begin{aligned} A_k(a,b,c,y)=\frac{\left( 1-aq^{k-1}\right) \left( 1-bq^{k-1}\right) }{\left( 1-q^k\right) \left( 1-cq^{k-1}\right) }D_yA_{k-1}(a,b,c,y). \end{aligned}$$
(1.14)

Iterating, we have

$$\begin{aligned} A_k(a,b,c,y)=\frac{(a,b;q)_k}{(q,c;q)_k}D_y^kA_0(a,b,c,y). \end{aligned}$$
(1.15)

Letting \(f(a,b,c,0,y)=A_0(a,b,c,y)=\sum _{n=0}^\infty \mu _n y^n\), we have

$$\begin{aligned} A_k(a,b,c,y)=\frac{(a,b;q)_k}{(q,c;q)_k}\sum _{n=0}^\infty \mu _n \frac{(q;q)_n}{(q;q)_{n-k}}y^{n-k}. \end{aligned}$$
(1.16)

By Eq. (1.12), we have

$$\begin{aligned}&f(a,b,c,x,y)&=\sum _{k=0}^\infty \frac{(a,b;q)_k}{(q,c;q)_k}\sum _{n=0}^\infty \mu _n \frac{(q;q)_n}{(q;q)_{n-k}}y^{n-k} x^k\\&=\sum _{n=0}^\infty \mu _n\sum _{k=0}^n\left[ \begin{array}{l} n \\ k \\ \end{array}\right] \frac{(a,b;q)_k}{(q,c;q)_k}x^ky^{n-k}=\sum _{n=0}^\infty \mu _n \phi _n^{(a,b,c)}(x,y\arrowvert q). \end{aligned}$$

On the other hand, if f(a, b, c, x, y) can be expanded in terms of \(\phi _n^{(a,b,c)}(x,y\arrowvert q)\), we can verify that f(a, b, c, x, y) satisfies Eq. (1.10). The proof of Eq. (1.10) is complete. Similarly, we can deduce Eq. (1.11). The proof of Theorem 2 is complete. \(\square \)

This paper is organized as follows. In Sect. 2, we generalize two generating functions for Andrews–Askey polynomials. In Sect. 3, we deduce generalizations of Ramanujan type q-beta integrals. In Sect. 4, we generalize q-Chu–Vandermonde formula.

2 Two generating functions for generalized Al-Salam–Carlitz polynomials

In this section, we generalize generating functions for Al-Salam–Carlitz polynomials.

Theorem 6

We have

$$\begin{aligned} \sum _{n=0}^\infty \phi _n^{(a,b,c)}(x,y\arrowvert q)\frac{\bigl (s/r;q\bigr )_nr^n}{(q;q)_n}&=\frac{(sy;q)_\infty }{(ry;q)_\infty }{}_3\phi _2\biggl [\begin{array}{l} \begin{array}{lll} a,b,s/r\\ c,sy \end{array} \end{array};q,rx\biggr ],\quad \nonumber \\&\qquad \max \{\left|rx\right|,\left|ry\right|\}<1,\end{aligned}$$
(2.1)
$$\begin{aligned} \sum _{n=0}^\infty \psi _n^{(a,b,c)}(x,y\arrowvert q)\frac{\bigl (r/s;q\bigr )_ns^n}{(q;q)_n}&=\frac{(ry;q)_\infty }{(sy;q)_\infty }{}_3\phi _2\biggl [\begin{array}{l} \begin{array}{lll} a,b,r/s\\ c,q/(sy) \end{array} \end{array};q,\frac{qx}{y}\biggr ],\quad \nonumber \\&\qquad \max \{\left|sy\right|,\left|qx/y\right|\}<1. \end{aligned}$$
(2.2)

Corollary 7

We have

$$\begin{aligned} \sum _{n=0}^\infty \phi _n^{(a,b,c)}(x,y\arrowvert q)(-1)^nq^{n\atopwithdelims ()2}\frac{s^n}{(q;q)_n}&=(sy;q)_\infty {}_2\phi _2\biggl [\begin{array}{l} \begin{array}{lll} a,b\\ c,sy \end{array} \end{array};q,sx\biggr ], \end{aligned}$$
(2.3)
$$\begin{aligned} \sum _{n=0}^\infty \psi _n^{(a,b,c)}(x,y\arrowvert q)\frac{s^n}{(q;q)_n}&=\frac{1}{(sy;q)_\infty }{}_3\phi _2\biggl [\begin{array}{l} \begin{array}{lll} a,b,0\\ c,q/(sy) \end{array} \end{array};q,\frac{qx}{y}\biggr ],\quad \nonumber \\&\qquad \max \{\left|sy\right|,\left|qx/y\right|\}<1. \end{aligned}$$
(2.4)

Remark 8

For \(r=0\) in Theorem 6, Eqs. (2.1) and (2.2) reduce to Eqs. (2.3) and (2.4), respectively. For \(s=0\) in Theorem 6, Eqs. (2.1) and (2.2) reduce to Eqs. (1.6) and (1.7), respectively.

Proof of Theorem 6

By the Weierstrass M-test, series \(\sum _{n=0}^\infty M_n\) is convergent when \(\lim _{n\rightarrow \infty }\left|\frac{M_{n+1}}{M_n}\right|<1\). We check that both sides of Eq. (2.1) are convergent if \(\max \{\left|rx\right|,\left|ry\right|\}<1\), that is,

$$\begin{aligned} \lim _{n\rightarrow \infty }\left|\frac{\phi _{n+1}^{(a,b,c)}(x,y\arrowvert q)(s/r;q)_{n+1}r^{n+1}/(q;q)_{n+1}}{\phi _n^{(a,b,c)}(x,y\arrowvert q)(s/r;q)_nr^n/(q;q)_n}\right|&=\left|ry\right|<1,\\ \lim _{n\rightarrow \infty }\left|\frac{(a,b,s/r;q)_{n+1}(rx)^{n+1}/(q,c,sy;q)_{n+1}}{(a,b,s/r;q)_n(rx)^n/(q,c,sy;q)_n}\right|&=\left|rx\right|<1. \end{aligned}$$

We denote the right-hand side of Eq. (2.1) by f(a, b, c, x, y), we can verify that f(a, b, c, x, y) satisfies Eq. (1.10), so we have

$$\begin{aligned} f(a,b,c,x,y)=\sum _{n=0}^\infty \mu _n \phi _n^{(a,b,c)}(x,y\arrowvert q) \end{aligned}$$
(2.5)

and

$$\begin{aligned} f(a,b,c,0,y)=\sum _{n=0}^\infty \mu _n y^n=\frac{(sy;q)_\infty }{(ry;q)_\infty }=\sum _{n=0}^\infty \frac{\bigl (s/r;q\bigr )_n(yr)^n}{(q;q)_n}. \end{aligned}$$
(2.6)

So f(a, b, c, x, y) is equal to the left-hand side of (2.1). Similarly, we can obtain Eq. (2.2). The proof is complete. \(\square \)

3 Generalizations of two of Ramanujan’s integrals

The following two integrals of Ramanujan [3] are quite famous.

Proposition 9

([3, Eqs. (2) and (3)]) For \(0<q=\exp (-2k^2)<1\) and \(m\in \mathbb R\). Suppose that \(\left|abq\right|<1\), we have

$$\begin{aligned} \int _{-\infty }^{\infty } \frac{e^{-x^2+2mx}}{\left( aq^{1/2}e^{2ikx},bq^{1/2}e^{-2ikx};q\right) _\infty }{{\mathrm{\mathrm{d}}}}x=\sqrt{\pi }e^{m^2}\frac{\left( -aqe^{2mki},-bqe^{-2mki};q\right) _\infty }{(abq;q)_\infty }. \end{aligned}$$
(3.1)

Suppose that \(\max \left\{ \left|aq^{1/2}e^{2mk}\right|,\left|bq^{1/2}e^{-2mk}\right|\right\} <1\), we have

$$\begin{aligned} \int _{-\infty }^{\infty } e^{-x^2+2mx}\left( -aqe^{2kx},-bqe^{-2kx};q\right) _\infty {{\mathrm{\mathrm{d}}}}x\nonumber \\=\sqrt{\pi }e^{m^2}\frac{(abq;q)_\infty }{\left( aq^{1/2}e^{2mk},bq^{1/2}e^{-2mk};q\right) _\infty }. \end{aligned}$$
(3.2)

Derivations of (3.1) and (3.2) for real values of the parameter m have been deduced by Askey [3]. Later on it became clear that these integrals are in fact valid for arbitrary complex values of the parameter m and they are thus instances of the standard Fourier transform with the exponential kernel by Atakishiyev and Feinsilver [5].

In this section, we have the following generalization of Ramanujan’s integrals.

Theorem 10

For \(m\in \mathbb R\), \(0<q=\exp (-2k^2)<1\). Suppose that \(\max \{\left|abq\right|,\left|qc/a\right|\}<1\), we have

$$\begin{aligned}&\int _{-\infty }^{\infty } \frac{e^{-x^2+2mx}}{\Bigl (aq^{1/2}e^{2ikx},bq^{1/2}e^{-2ikx};q\Bigr )_\infty }{}_3\phi _2\left[ \begin{array}{l} \begin{array}{ccc} r,s,0\\ t,q^{1/2}e^{-2ikx}/a \end{array} \end{array};q,\frac{qc}{a}\right] {{\mathrm{\mathrm{d}}}}x\nonumber \\&\quad =\sqrt{\pi }e^{m^2}\frac{\Bigl (-aqe^{2mki},-bqe^{-2mki};q\Bigr )_\infty }{(abq;q)_\infty }{}_3\phi _2\left[ \begin{array}{l} \begin{array}{ccc} r,s,-e^{2mki}/b\\ t,1/(ab) \end{array} \end{array};q,\frac{qc}{a}\right] .\quad \quad \quad \end{aligned}$$
(3.3)

Suppose that \(\max \left\{ \left|aq^{1/2}e^{2mk}\right|,\left|bq^{1/2}e^{-2mk}\right|,\left|cq^{1/2}e^{2mk}\right|\right\} <1\), we have

$$\begin{aligned}&\int _{-\infty }^{\infty } e^{-x^2+2mx}\Bigl (-aqe^{2kx},-bqe^{-2kx};q\Bigr )_\infty {}_2\phi _2\left[ \begin{array}{l} \begin{array}{ccc} r,s\\ t,-aqe^{2kx} \end{array} \end{array};q,-cqe^{2kx}\right] {{\mathrm{\mathrm{d}}}}x\nonumber \\&\quad =\sqrt{\pi }e^{m^2}\frac{(abq;q)_\infty }{\Bigl (aq^{1/2}e^{2mk},bq^{1/2}e^{-2mk};q\Bigr )_\infty }{}_3\phi _2\left[ \begin{array}{l} \begin{array}{lll} r,s,bq^{1/2}e^{-2mk}\\ t,abq \end{array} \end{array};q,cq^{1/2}e^{2mk}\right] .\nonumber \\ \end{aligned}$$
(3.4)

Remark 11

For \(c=0\) in Theorem 10, Eqs. (3.3) and (3.4) reduce to Eqs. (3.1) and (3.2), respectively.

Proof of Theorem 10

It is easily seen that

$$\begin{aligned} \left( \left|q^{\frac{1}{2}}/a\right|;q\right) _n\le \left|\left( q^{\frac{1}{2}}e^{-2ikx}/a;q\right) _n\right| \le \left( -\left|q^{\frac{1}{2}}/a\right|;q\right) _n \end{aligned}$$
(3.5)

and

$$\begin{aligned} \sum _{n=0}^\infty \frac{\bigl (\left|r\right|,\left|s\right|;q\bigr )_n}{\Bigl (-\left|q\right|,-\left|t\right|,-\left|q^{1/2}/a\right|;q\Bigr )_n}\left|\frac{qc}{a}\right|^n\le & {} \left| {}_3\phi _2\biggl [\begin{array}{l} \begin{array}{ccc} r,s,0\\ t,q^{1/2}e^{-2ikx}/a \end{array} \end{array};q,\frac{qc}{a}\biggr ] \right|\nonumber \\\le & {} \sum _{n=0}^\infty \frac{\bigl (-\left|r\right|,-\left|s\right|;q\bigr )_n}{\Bigl (\left|q\right|,\left|t\right|,\left|q^{1/2}/a\right|;q\Bigr )_n}\left|\frac{qc}{a}\right|^n.\nonumber \\ \end{aligned}$$
(3.6)

Thus, we have

$$\begin{aligned}&\left|\int _{-\infty }^{\infty } \frac{e^{-x^2+2mx}}{\Bigl (aq^{1/2}e^{2ikx},bq^{1/2}e^{-2ikx};q\Bigr )_\infty }{{\mathrm{\mathrm{d}}}}x\right|\cdot \sum _{n=0}^\infty \frac{\bigl (\left|r\right|,\left|s\right|;q\bigr )_n}{\Bigl (-\left|q\right|,-\left|t\right|,-\left|q^{1/2}/a\right|;q\Bigr )_n}\left|\frac{qc}{a}\right|^n\nonumber \\&\quad \le \left|\int _{-\infty }^{\infty }\frac{e^{-x^2+2mx}}{\Bigl (aq^{1/2}e^{2ikx},bq^{1/2}e^{-2ikx};q\Bigr )_\infty }{}_3\phi _2\biggl [\begin{array}{l} \begin{array}{lll} r,s,0\\ t,q^{1/2}e^{-2ikx}/a \end{array} \end{array};q,\frac{qc}{a}\biggr ]{{\mathrm{\mathrm{d}}}}x\right|\nonumber \\&\quad \le \left|\int _{-\infty }^{\infty } \frac{e^{-x^2+2mx}}{\Bigl (aq^{1/2}e^{2ikx},bq^{1/2}e^{-2ikx};q\Bigr )_\infty }{{\mathrm{\mathrm{d}}}}x\right| \cdot \sum _{n=0}^\infty \frac{\bigl (-\left|r\right|,-\left|s\right|;q\bigr )_n}{\Bigl (\left|q\right|,\left|t\right|,\left|q^{1/2}/a\right|;q\Bigr )_n}\left|\frac{qc}{a}\right|^n.\nonumber \\ \end{aligned}$$
(3.7)

Denoting the right-hand side of Eq. (3.3) by f(r, s, t, c, a) and utilizing Eqs. (3.1) and (3.7), we have

$$\begin{aligned} \left|f(r,s,t,c,a)\right|&\le \left|\sqrt{\pi }e^{m^2}\frac{\Bigl (-aqe^{2mki},-bqe^{-2mki};q\Bigr )_\infty }{(abq;q)_\infty }\right|\nonumber \\&\quad \times \sum _{n=0}^\infty \frac{\bigl (-\left|r\right|,-\left|s\right|;q\bigr )_n}{\Bigl (\left|q\right|,\left|t\right|,\left|q^{1/2}/a\right|;q\Bigr )_n}\left|\frac{qc}{a}\right|^n\nonumber \\&\le \sqrt{\pi }e^{m^2}\frac{\Bigl (-\left|aq\right|,-\left|bq\right|;q\Bigr )_\infty }{(\left|abq\right|;q)_\infty }\nonumber \\&\quad \times \sum _{n=0}^\infty \frac{\bigl (-\left|r\right|,-\left|s\right|;q\bigr )_n}{\Bigl (\left|q\right|,\left|t\right|,\left|q^{1/2}/a\right|;q\Bigr )_n}\left|\frac{qc}{a}\right|^n. \end{aligned}$$
(3.8)

From the Weierstrass M-test, we know that for \(\max \{\left|abq\right|,\left|qc/a\right|\}<1\), the function f(r, s, t, c, a) is uniformly absolutely convergent, so f(r, s, t, c, a) is an analytic function of r, s, t, c and a for \(\max \{\left|abq\right|,\left|qc/a\right|\}<1\) (see also [17], p. 516]). Thus f(r, s, t, c, a) is analytic near \((r,s,t,c,a)=(0,0,0,0,0)\) (see also [4], p. 220] and [17], p. 511]). We can check that f(r, s, t, c, a) satisfies Eq. (1.11), so the left-hand side of Eq. (3.3) equals

$$\begin{aligned} f(r,s,t,c,a)=\sum _{n=0}^\infty \mu _n \psi _n^{(r,s,t)}(c,a\arrowvert q), \end{aligned}$$
(3.9)

where

$$\begin{aligned} f(r,s,t,0,a)&=\sum _{n=0}^\infty \mu _n a^n=\sqrt{\pi }e^{m^2}\frac{\Bigl (-aqe^{2mki},-bqe^{-2mki};q\Bigr )_\infty }{(abq;q)_\infty }\quad by\, (3.1) \\&=\int _{-\infty }^{\infty } \frac{e^{-x^2+2mx}}{\Bigl (aq^{1/2}e^{2ikx},bq^{1/2}e^{-2ikx};q\Bigr )_\infty }{{\mathrm{\mathrm{d}}}}x\\&=\int _{-\infty }^{\infty } \frac{e^{-x^2+2mx}}{\Bigl (bq^{1/2}e^{-2ikx};q\Bigr )_\infty }\left\{ \sum _{n=0}^\infty \frac{\bigl (aq^{1/2}e^{2ikx}\bigr )^n}{(q;q)_n}\right\} {{\mathrm{\mathrm{d}}}}x. \end{aligned}$$

So we have

$$\begin{aligned} f(r,s,t,c,a)=\int _{-\infty }^{\infty } \frac{e^{-x^2+2mx}}{\Bigl (bq^{1/2}e^{-2ikx};q\Bigr )_\infty }\left\{ \sum _{n=0}^\infty \psi _n^{(r,s,t)}(c,a\arrowvert q)\frac{\bigl (q^{1/2}e^{2ikx}\bigr )^n}{(q;q)_n}\right\} {{\mathrm{\mathrm{d}}}}x, \end{aligned}$$
(3.10)

which is equal to the left-hand side of Eq. (3.3) by Eq. (2.4). Similarly, we can gain Eq. (3.4). The proof of Theorem 10 is complete. \(\square \)

4 Generalizations of q-Chu–Vandermonde formula

The q-Chu–Vandermonde formula is [9, Eq. (II.6)]

$$\begin{aligned} {}_2\phi _1\biggl [\begin{array}{l} q^{-n}, a \\ c \end{array} ;q,q\biggr ]=\frac{\bigl (c/a;q\bigr )_n}{(c;q)_n}a^n. \end{aligned}$$
(4.1)

In this section, we now extend the q-Chu–Vandermonde formula.

Theorem 12

For \(n\in \mathbb N_0\), we have

$$\begin{aligned}&\sum _{k=0}^n\frac{\bigl (q^{-n},a;q\bigr )_kq^k}{(q,cd;q)_k}{}_3\phi _2\biggl [\begin{array}{l} \begin{array}{ccc} r,s,aq^k\\ t,qa/(cd) \end{array} \end{array};q,\frac{qg}{d}\biggr ]&\nonumber \\&\quad =\frac{\bigl (cd/a;q\bigr )_n a^n}{(cd;q)_n}{}_3\phi _2\biggl [\begin{array}{l} \begin{array}{ccc} r,s,a\\ t,aq^{1-n}/(cd) \end{array} \end{array};q,\frac{qg}{d}\biggr ],\quad \left|qg/d\right|<1,\end{aligned}$$
(4.2)
$$\begin{aligned}&\sum _{k=0}^n\frac{\bigl (q^{-n},a;q\bigr )_kq^k}{(q,cd;q)_k}{}_3s\phi _2\biggl [\begin{array}{l} \begin{array}{ccc} r,s,aq^k\\ t,cdq^k \end{array} \end{array};q,\frac{cg}{a}\biggr ]&\nonumber \\&\quad =\frac{\bigl (cd/a;q\bigr )_n a^n}{(cd;q)_n}{}_3\phi _2\biggl [\begin{array}{l} \begin{array}{ccc} r,s,aq^n\\ t,cdq^n \end{array} \end{array};q,\frac{cgq^n}{a}\biggr ],\quad \left|cg/a\right|<1. \end{aligned}$$
(4.3)

Remark 13

For \(g=0\) in Theorem 12, Eqs. (4.2) and (4.3) reduce to (4.1), respectively.

Proof of Theorem 12

First, we can rewrite Eq. (4.1) equivalently by

$$\begin{aligned} \sum _{k=0}^n\frac{\bigl (q^{-n},a;q\bigr )_kq^k}{(q;q)_k} \frac{\bigl (cdq^k;q\bigr )_\infty }{\bigl (cd/a;q\bigr )_\infty }=a^n\frac{\bigl (cdq^n;q\bigr )_\infty }{\bigl (cdq^n/a;q\bigr )_\infty }. \end{aligned}$$
(4.4)

We denote the right-hand side of (4.2) by F(r, s, t, g, d), we can check that F(r, s, t, g, d) satisfies Eq. (1.11). By Eq. (4.4), we have

$$\begin{aligned} F(r,s,t,g,d)=\sum _{j=0}^\infty \mu _j \psi _j^{(r,s,t)}(g,d\arrowvert q) \end{aligned}$$
(4.5)

and

$$\begin{aligned} F(r,s,t,0,d)&=\sum _{j=0}^\infty \mu _j d^j=a^n\frac{\bigl (cdq^n;q\bigr )_\infty }{\bigl (cdq^n/a;q\bigr )_\infty }\nonumber \\&=\sum _{k=0}^n\frac{\bigl (q^{-n},a;q\bigr )_kq^k}{(q;q)_k}\sum _{j=0}^\infty \frac{\bigl (aq^k;q\bigr )_j\bigl (cd/a\bigr )^j}{(q;q)_j}. \end{aligned}$$

So we have

$$\begin{aligned} F(r,s,t,g,d)=\sum _{k=0}^n\frac{\bigl (q^{-n},a;q\bigr )_kq^k}{(q;q)_k} \sum _{j=0}^\infty \frac{\bigl (aq^k;q\bigr )_j\bigl (c/a\bigr )^j}{(q;q)_j} \psi _j^{(r,s,t)}(g,d\arrowvert q), \end{aligned}$$
(4.6)

which is equal to the left-hand side of (4.2) by Eq. (2.2). Similarly, we can deduce Eq. (4.3). The proof is complete. \(\square \)