A Note on Generalized q-Difference Equations and Their Applications Involving q-Hypergeometric Functions

Abstract

Our investigation is motivated essentially by the demonstrated applications of the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, in many diverse areas. Here, in this paper, we use two q-operators $\mathbb{T}(a,b,c,d,e,y{D}_x)$ and $\mathbb{E}(a,b,c,d,e,y{\theta }_x)$ to derive two potentially useful generalizations of the q-binomial theorem, a set of two extensions of the q-Chu-Vandermonde summation formula and two new generalizations of the Andrews-Askey integral by means of the q-difference equations. We also briefly describe relevant connections of various special cases and consequences of our main results with a number of known results.

1. Introduction, Definitions and Preliminaries

Throughout this paper, we refer to [1] for definitions and notations. We also suppose that $0<q<1$. For complex numbers a, the q-shifted factorials are defined by

$$\begin{array}{c}\hfill {(a;q)}_0:=1,\end{array}$$

(1)

$$\begin{array}{c}\hfill {(a;q)}_n:=\prod _{k=0}^{n-1}(1-a{q}^k)=(1-a)(1-aq)(1-a{q}^2)\cdots (1-a{q}^{n-1})(n\in \mathbb{N}:=\{1,2,3,\cdots \})\end{array}$$

(2)

and

$$\begin{array}{c}\hfill {(a;q)}_{\infty }:=\prod _{k=0}^{\infty }(1-a{q}^k),\end{array}$$

(3)

where (see, for example, [1,2])

$${(a;q)}_n=\frac{{(a;q)}_{\infty }}{{(a{q}^n;q)}_{\infty }}\mathrm{and}{(a;q)}_{n+m}={(a;q)}_n{(a{q}^n;q)}_m$$

and

$${\left(\frac{q}{a};q\right)}_n={(-a)}^{-n}{q}^{\left({}_2^{n+1}\right)}\frac{{(a{q}^{-n};q)}_{\infty }}{{(a;q)}_{\infty }}.$$

We adopt the following notation:

$${(a_1,a_2,\cdots ,a_r;q)}_m={(a_1;q)}_m{(a_2;q)}_m\cdots {(a_r;q)}_m(m\in \mathbb{N}:=\{1,2,3,\cdots \}).$$

Also, for m large, we have

$${(a_1,a_2,\cdots ,a_r;q)}_{\infty }={(a_1;q)}_{\infty }{(a_2;q)}_{\infty }\cdots {(a_r;q)}_{\infty }.$$

The q-binomial coefficient is defined by

$${\left[\begin{array}{c}n\\ k\end{array}\right]}_q:=\frac{{(q;q)}_n}{{(q;q)}_k{(q;q)}_{n-k}}.$$

(4)

The basic (or q-) hypergeometric function of the variable z and with $\mathfrak{r}$ numerator and $\mathfrak{s}$ denominator parameters is defined as follows (see, for details, the monographs by Slater ([2], Chapter 3) and by Srivastava and Karlsson ([3], p. 347, equation (272)); see also [4,5,6]):

$${}_{\mathfrak{r}}{\Phi }_{\mathfrak{s}}\left[\begin{array}{c}\hfill a_1,a_2,\cdots ,a_{\mathfrak{r}};\\ \\ \hfill b_1,b_2,\cdots ,b_{\mathfrak{s}};\end{array}q;z\right]:=\sum _{n=0}^{\infty }{\left[{(-1)}^n{q}^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}\right]}^{1+\mathfrak{s}-\mathfrak{r}}\frac{{(a_1,a_2,\cdots ,a_{\mathfrak{r}};q)}_n}{{(b_1,b_2,\cdots ,b_{\mathfrak{s}};q)}_n}\frac{z^n}{{(q;q)}_n},$$

where $q\ne 0$ when $\mathfrak{r}>\mathfrak{s}+1$. We also note that

$${}_{\mathfrak{r}+1}{\Phi }_{\mathfrak{r}}\left[\begin{array}{c}\hfill a_1,a_2,\cdots ,a_{\mathfrak{r}+1}\\ \\ \hfill b_1,b_2,\cdots ,b_{\mathfrak{r}};\end{array}q;z\right]=\sum _{n=0}^{\infty }\frac{{(a_1,a_2,\cdots ,a_{\mathfrak{r}+1};q)}_n}{{(b_1,b_2,\cdots ,b_{\mathfrak{r}};q)}_n}\frac{z^n}{{(q;q)}_n}.$$

We remark in passing that, in a recently-published survey-cum-expository review article, the so-called $(p,q)$-calculus was exposed to be a rather trivial and inconsequential variation of the classical q-calculus, the additional parameter p being redundant or superfluous (see, for details, ([7], p. 340)).

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several areas of Number Theory such as the Theory of Partitions and are useful also in a wide variety of fields including, for example, Combinatorial Analysis, Finite Vector Spaces, Lie Theory, Particle Physics, Non-Linear Electric Circuit Theory (see [8,9]), Mechanical Engineering (see [10]), Theory of Heat Conduction, Quantum Mechanics, Cosmology, Computation of Fractional-Order Derivatives (see [11]) and Statistics [see also ([3], pp. 350–351), and the references cited therein]. Here, in our present investigation, we are mainly concerned with the Cauchy polynomials $p_n(x,y)$ as given below (see [1,12]):

$$p_n(x,y):=(x-y)(x-qy)\cdots (x-q^{n-1}y)={\left(\frac{y}{x};q\right)}_n{x}^n,$$

(5)

together with the following Srivastava-Agarwal type generating function (see also [13]):

$$\sum _{n=0}^{\infty }{p}_n(x,y)\frac{{(\lambda ;q)}_n{t}^n}{{(q;q)}_n}={}_2{\Phi }_1\left[\begin{array}{c}\hfill \lambda ,\frac{y}{x};\\ \\ \hfill 0;\end{array}q;xt\right].$$

(6)

In particular, for $\lambda =0$ in (6), we get the following simpler generating function [12]:

$$\sum _{n=0}^{\infty }{p}_n(x,y)\frac{t^n}{{(q;q)}_n}=\frac{{(yt;q)}_{\infty }}{{(xt;q)}_{\infty }}.$$

(7)

The generating function (7) is also the homogeneous version of the Cauchy identity or the following q-binomial theorem (see, for example, [1,2,3,14]):

$$\sum _{k=0}^{\infty }\frac{{(a;q)}_k}{{(q;q)}_k}{z}^k={}_1{\Phi }_0\left[\begin{array}{c}\hfill a;\\ \\ \hfill \overline{};\end{array}q;z\right]=\frac{{(az;q)}_{\infty }}{{(z;q)}_{\infty }}\left(\right|z|<1).$$

(8)

Upon further setting $a=0$, this last relation (8) becomes Euler’s identity (see, for example, [1]):

$$\sum _{k=0}^{\infty }\frac{z^k}{{(q;q)}_k}=\frac{1}{{(z;q)}_{\infty }}\left(\right|z|<1)$$

(9)

and its inverse relation given below [1]:

$$\sum _{k=0}^{\infty }\frac{{(-1)}^k}{{(q;q)}_k}{q}^{\left(\genfrac{}{}{0pt}{}{k}{2}\right)}{z}^k={(z;q)}_{\infty }.$$

(10)

Based upon the q-binomial theorem (8) and Heine’s transformations, Srivastava et al. [15] established a set of two presumably new theta-function identities (see, for details, [15]).

The following usual q-difference operators are defined by [16,17,18]

$$D_a\left\{f\left(a\right)\right\}:=\frac{f\left(a\right)-f\left(qa\right)}{a},{\theta }_a=\left\{f\left(a\right)\right\}:=\frac{f\left(q^{-1}a\right)-f\left(a\right)}{q^{-1}a},$$

(11)

and their Leibniz rules are given by (see [19])

$$\begin{array}{c}\hfill D_a^n\left\{f\left(a\right)g\left(a\right)\right\}=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{q}^{k(k-n)}{D}_a^k\left\{f\left(a\right)\right\}{D}_a^{n-k}\left\{g\left(q^{k}a\right)\right\}\end{array}$$

(12)

and

$$\begin{array}{c}\hfill {\theta }_a^n\left\{f\left(a\right)g\left(a\right)\right\}=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\theta }_a^k\left\{f\left(a\right)\right\}{\theta }_a^{n-k}\left\{g\left(q^{-k}a\right)\right\},\end{array}$$

(13)

respectively. Here, and in what follows, $D_a^0$ and ${\theta }_a^0$ are understood as the identity operators.

Recently, Chen and Liu [16,20] constructed the following pair of augmentation operators, which is of great significance for deriving identities by applying its various special cases:

$$\mathbb{T}\left(b{D}_x\right)=\sum _{n=0}^{\infty }\frac{{\left(b{D}_x\right)}^n}{{(q;q)}_n}\mathrm{and}\mathbb{E}\left(b{\theta }_x\right)=\sum _{n=0}^{\infty }\frac{{\left(b{\theta }_x\right)}^n}{{(q;q)}_n}.$$

(14)

Subsequently, Chen and Gu [21] defined the Cauchy augmentation operators as follows:

$$\mathbb{T}(a,b{D}_x)=\sum _{n=0}^{\infty }\frac{{(a;q)}_n}{{(q;q)}_n}{\left(b{D}_x\right)}^n$$

(15)

and

$$\mathbb{E}(a,b{\theta }_x)=\sum _{n=0}^{\infty }\frac{{(b;q)}_n}{{(q;q)}_n}{(-b{\theta }_x)}^n.$$

(16)

On the other hand, Fang [22] and Zhang and Wang [23] considered the following finite generalized q-exponential operators with two parameters:

$$\mathbb{T}\left[\begin{array}{c}{q}^{-N},w\\ v\end{array}|q;t{D}_x\right]=\sum _{n=0}^N\frac{{(q^{-N},w;q)}_n}{{(v,q;q)}_n}{\left(t{D}_x\right)}^n$$

(17)

and

$$\mathbb{E}\left[\begin{array}{c}{q}^{-N},w\\ v\end{array}|q;t{\theta }_x\right]=\sum _{n=0}^N\frac{{(q^{-N},w;q)}_n}{{(v,q;q)}_n}{\left(t{\theta }_x\right)}^n.$$

(18)

Moreover, Li and Tan [24] constructed two generalized q-exponential operators with three parameters as follows:

$$\mathbb{T}\left[\begin{array}{c}u,v\\ w\end{array}|q;t{D}_x\right]=\sum _{n=0}^{\infty }\frac{{(u,v;q)}_n}{{(w,q;q)}_n}{\left(t{D}_x\right)}^n$$

(19)

and

$$\mathbb{E}\left[\begin{array}{c}u,v\\ w\end{array}|q;t{\theta }_x\right]=\sum _{n=0}^{\infty }\frac{{(u,v;q)}_n}{{(w,q;q)}_n}{\left(t{\theta }_x\right)}^n.$$

(20)

Finally, we recall that Cao et al. [25] constructed the following q-operators:

$$\begin{array}{c}\hfill \mathbb{T}(a,b,c,d,e,y{D}_x)=\sum _{n=0}^{\infty }\frac{{(a,b,c;q)}_n}{{(q,d,e;q)}_n}{\left(y{D}_x\right)}^n\end{array}$$

(21)

and

$$\begin{array}{c}\hfill \mathbb{E}(a,b,c,d,e,y{\theta }_x)=\sum _{n=0}^{\infty }\frac{{(-1)}^n{q}^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}{(a,b,c;q)}_n}{{(q,d,e;q)}_n}{\left(y{\theta }_x\right)}^n\end{array}$$

(22)

and thereby generalized Arjika’s results in [26] by using the q-difference equations (see, for details, [25]).

We remark that the q-operator (21) is a particular case of the homogeneous q-difference operator $\mathbb{T}(\mathbf{a},\mathbf{b},c{D}_x)$ (see [27]) by taking

$$\mathbf{a}=(a,b,c),\mathbf{b}=(d,e)\mathrm{and}c=y.$$

Furthermore, for $b=c=d=e=0$, the q-operator (22) reduces to the operator $\tilde{L}(a,y;{\theta }_x)$ which was investigated by Srivastava et al. [28].

Cao et al. [25] used the q-operators (21) and (22) and gave the following results.

Proposition 1.

(see ([25], theorem 3)) Let f (a, b, c, d, e, x, y) be a seven-variable analytic function in a neighborhood of (a, b, c, d, e, x, y) = (0, 0, 0, 0, 0, 0, 0)$\in {\mathbb{C}}^7$.

(I) 

If $f(a,b,c,d,e,x,y)$ satisfies the following difference equation:

$$\begin{array}{cc}\hfill & x\left\{f\right(a,b,c,d,e,x,y)-f(a,b,c,d,e,x,yq)\hfill \\ \hfill & -(d+e)q^{-1}[f(a,b,c,d,e,x,yq)-f(a,b,c,d,e,x,y{q}^2)]\hfill \\ \hfill & +de{q}^{-2}[f(a,b,c,d,e,x,y{q}^2)-f(a,b,c,d,e,x,y{q}^3)]\}\hfill \\ \hfill & =y\left\{\right[f(a,b,c,d,e,x,y)-f(a,b,c,d,e,xq,y)]\hfill \\ \hfill & -(a+b+c)\left[f\right(a,b,c,d,e,x,yq)-f(a,b,c,d,e,xq,yq\left)\right]\hfill \\ \hfill & +(ab+ac+bc)[f(a,b,c,d,e,x,y{q}^2)-f(a,b,c,d,e,xq,y{q}^2)]\hfill \\ \hfill & -abc[f(a,b,c,d,e,x,y{q}^3)-f(a,b,c,d,e,xq,y{q}^3)]\},\hfill \end{array}$$

(23)

then

$$\begin{array}{c}\hfill f(a,b,c,d,e,x,y)=\mathbb{T}(a,b,c,d,e,y{D}_x)\left\{f(a,b,c,d,e,x,0)\right\}.\end{array}$$

(24)

(II) 

If $f(a,b,c,d,e,x,y)$ satisfies the following difference equation:

$$\begin{array}{cc}\hfill & x\left\{f\right(a,b,c,d,e,xq,y)-f(a,b,c,d,e,xq,yq)\hfill \\ \hfill & -(d+e)q^{-1}[f(a,b,c,d,e,xq,yq)-f(a,b,c,d,e,xq,y{q}^2)]\hfill \\ \hfill & +de{q}^{-2}[f(a,b,c,d,e,xq,y{q}^2)-f(a,b,c,d,e,xq,y{q}^3)]\}\hfill \\ \hfill & =y\left\{\right[f(a,b,c,d,e,xq,yq)-f(a,b,c,d,e,x,yq)]\hfill \\ \hfill & -(a+b+c)[f(a,b,c,d,e,xq,y{q}^2)-f(a,b,c,d,e,x,y{q}^2)]\hfill \\ \hfill & +(ab+ac+bc)[f(a,b,c,d,e,xq,y{q}^3)-f(a,b,c,d,e,x,y{q}^3)]\hfill \\ \hfill & -abc[f(a,b,c,d,e,xq,y{q}^4)-f(a,b,c,d,e,x,y{q}^4)]\},\hfill \end{array}$$

(25)

then

$$\begin{array}{c}\hfill f(a,b,c,d,e,x,y)=\mathbb{E}(a,b,c,d,e,y{\theta }_x)\left\{f(a,b,c,d,e,x,0)\right\}.\end{array}$$

(26)

Liu [29,30] initiated the method based upon q-difference equations and deduced several results involving Bailey’s ${}_6{\psi }_6,q$-Mehler formulas for the Rogers-Szegö polynomials and q-integral version of the Sears transformation.

Lemma 1.

Each of the following q-identities holds true:

$$\begin{array}{c}\hfill D_a^k\left\{\frac{1}{{(as;q)}_{\infty }}\right\}=\frac{s^k}{{(as;q)}_{\infty }},\end{array}$$

(27)

$$\begin{array}{c}\hfill {\theta }_a^k\left\{\frac{1}{{(as;q)}_{\infty }}\right\}=\frac{s^k{q}^{-\left(\genfrac{}{}{0pt}{}{k}{2}\right)}}{{(as{q}^{-k};q)}_{\infty }},\end{array}$$

(28)

$$\begin{array}{c}\hfill D_a^k\left\{{(as;q)}_{\infty }\right\}={(-s)}^k{q}^{\left(\genfrac{}{}{0pt}{}{k}{2}\right)}{(as{q}^k;q)}_{\infty },\end{array}$$

(29)

$$\begin{array}{c}\hfill {\theta }_a^k\left\{{(as;q)}_{\infty }\right\}={(-s)}^k{(as;q)}_{\infty },\end{array}$$

(30)

$$\begin{array}{c}\hfill D_a^n\left\{\frac{{(as;q)}_{\infty }}{{(a\omega ;q)}_{\infty }}\right\}={\omega }^n\frac{{\left(\frac{s}{\omega };q\right)}_n}{{(as;q)}_n}\frac{{(as;q)}_{\infty }}{{(a\omega ;q)}_{\infty }}\end{array}$$

(31)

and

$$\begin{array}{c}\hfill {\theta }_a^n\left\{\frac{{(as;q)}_{\infty }}{{(a\omega ;q)}_{\infty }}\right\}={\left(-\frac{q}{a}\right)}^n\frac{{\left(\frac{s}{\omega };q\right)}_n}{{\left(\frac{q}{a\omega };q\right)}_n}\frac{{(as;q)}_{\infty }}{{(a\omega ;q)}_{\infty }}.\end{array}$$

(32)

We now state and prove the q-difference formulas as Theorem 1 below.

Theorem 1.

Each of the following assertions holds true:

$$\begin{array}{cc}\hfill & \mathbb{T}(r,f,g,v,w,u{D}_a)\left\{\frac{{(as;q)}_{\infty }}{{(az,at;q)}_{\infty }}\right\}\hfill \\ \hfill & =\frac{{(as;q)}_{\infty }}{{(az,at;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{\left(r,f,g,\frac{s}{z},at;q\right)}_k{\left(zu\right)}^k}{{(v,w,as,q;q)}_k}{}_3{\Phi }_2\left[\begin{array}{c}\begin{array}{c}\hfill r{q}^k,f{q}^k,g{q}^k;\\ \\ \hfill v{q}^k,w{q}^k;\end{array}\end{array}q;ut\right]\hfill \end{array}$$

(33)

and

$$\begin{array}{cc}\hfill & \mathbb{E}(r,f,g,v,w,u{\theta }_a)\left\{\frac{{(az,at;q)}_{\infty }}{{(as;q)}_{\infty }}\right\}\hfill \\ \hfill & =\frac{{(az,at;q)}_{\infty }}{{(as;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{\left(r,f,g,\frac{z}{s},\frac{q}{at};q\right)}_k{(-ut)}^k}{{\left(v,w,\frac{q}{as},q;q\right)}_k}{}_3{\Phi }_3\left[\begin{array}{c}\begin{array}{c}\hfill r{q}^k,f{q}^k,g{q}^k;\\ \\ \hfill v{q}^k,w{q}^k,0;\end{array}\end{array}q;-ut\right],\hfill \end{array}$$

(34)

provided that $max\left\{\left|az\right|,\left|as\right|,\left|at\right|,\left|ut\right|\right\}<1.$

Proof. 

By means of the definition (21) of the operator $\mathbb{T}(r,f,g,v,w,u{D}_a)$ and the Leibniz rule (12), we observe that

$$\begin{array}{cc}\hfill & \mathbb{T}(r,f,g,v,w,u{D}_a)\left\{\frac{{(as;q)}_{\infty }}{{(az,at;q)}_{\infty }}\right\}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\frac{{(r,f,g;q)}_n{u}^n}{{(v,w,q;q)}_n}{D}_a^n\left\{\frac{{(as;q)}_{\infty }}{{(az,at;q)}_{\infty }}\right\}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\frac{{(r,f,g;q)}_n{u}^n}{{(v,w,q;q)}_n}\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{q}^{k(k-n)}{D}_a^k\left\{\frac{{(as;q)}_{\infty }}{{(az;q)}_{\infty }}\right\}\hfill \\ \hfill & ·D_a^{n-k}\left\{\frac{1}{{(at{q}^k;q)}_{\infty }}\right\}.\hfill \end{array}$$

(35)

Now, using the q-identities (27) and (31), we find that

$$\begin{array}{cc}\hfill & \mathbb{T}(r,f,g,v,w,u{D}_a)\left\{\frac{{(as;q)}_{\infty }}{{(az,at;q)}_{\infty }}\right\}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\frac{{(r,f,g;q)}_n{u}^n}{{(v,w,q;q)}_n}\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{q}^{k(k-n)}\frac{{\left(\frac{s}{z};q\right)}_k{z}^k}{{(as;q)}_k}\frac{{(as;q)}_{\infty }}{{(az;q)}_{\infty }}\frac{{\left(t{q}^k\right)}^{n-k}}{{(at{q}^k;q)}_{\infty }}\hfill \\ \hfill & =\frac{{(as;q)}_{\infty }}{{(az,at;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{\left(\frac{s}{z},at;q\right)}_k{z}^k}{{(as,q;q)}_k}\sum _{n=k}^{\infty }\frac{{(r,f,g;q)}_n{u}^n{t}^{n-k}}{{(v,w;q)}_n{(q;q)}_{n-k}}.\hfill \end{array}$$

(36)

Upon setting $n-k=n$, the right-hand side of (36) takes the following form:

$$\frac{{(as;q)}_{\infty }}{{(az,at;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{\left(r,f,g,\frac{s}{z},at;q\right)}_k{\left(uz\right)}^k}{{(v,w,as,q;q)}_k}\sum _{n=0}^{\infty }\frac{{(r{q}^k,f{q}^k,g{q}^k;q)}_n{\left(ut\right)}^n}{{(v{q}^k,w{q}^k,q;q)}_n}.$$

(37)

Summarizing the above observations (35) to (37), we get the assertion (33) of Theorem 1.

Similarly, by means of the definition (22) of the operator $\mathbb{E}(r,f,g,v,w,u{\theta }_a)$ and the Leibniz rule (13), we get

$$\begin{array}{cc}\hfill & \mathbb{E}(r,f,g,v,w,u{\theta }_a)\left\{\frac{{(az,at;q)}_{\infty }}{{(as;q)}_{\infty }}\right\}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\frac{{(-1)}^n{q}^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}{(r,f,g;q)}_n{u}^n}{{(v,w,q;q)}_n}{\theta }_a^n\left\{\frac{{(az,at;q)}_{\infty }}{{(as;q)}_{\infty }}\right\}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\frac{{(-1)}^n{q}^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}{(r,f,g;q)}_n{u}^n}{{(v,w,q;q)}_n}\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\theta }_a^k\left\{\frac{{(az;q)}_{\infty }}{{(as;q)}_{\infty }}\right\}{\theta }_a^{n-k}\left\{{(at{q}^{-k};q)}_{\infty }\right\},\hfill \end{array}$$

(38)

By using the q-identities (30) and (32), we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}(r,f,g,v,w,u{\theta }_a)\left\{\frac{{(az,at;q)}_{\infty }}{{(as;q)}_{\infty }}\right\}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\frac{{(-1)}^n{q}^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}{(r,f,g;q)}_n{u}^n}{{(v,w,q;q)}_n}\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q\frac{{\left(-\frac{q}{a}\right)}^k{\left(\frac{z}{s};q\right)}_k}{{\left(\frac{q}{as};q\right)}_k}\hfill \\ \hfill & ·\frac{{(az;q)}_{\infty }}{{(as;q)}_{\infty }}{(at{q}^{-k};q)}_{\infty }{(-t{q}^{-k})}^{n-k}\hfill \\ \hfill & =\frac{{(az,at;q)}_{\infty }}{{(as;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{(-1)}^k{q}^{-\left(\genfrac{}{}{0pt}{}{k}{2}\right)}{\left(\frac{z}{s},\frac{q}{at};q\right)}_k{t}^k}{{\left(\frac{q}{as},q;q\right)}_k}\hfill \\ \hfill & ·\sum _{n=k}^{\infty }\frac{q^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)-k(n-k)}{(r,f,g;q)}_n{u}^n{t}^{n-k}}{{(v,w;q)}_n{(q;q)}_{n-k}}.\hfill \end{array}$$

(39)

Now, if we set $n-k=n$, the right-hand side of (39) becomes

$$\begin{array}{c}\hfill \frac{{(az,at;q)}_{\infty }}{{(as;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{\left(r,f,g,\frac{z}{s},\frac{q}{at};q\right)}_k{(-ut)}^k}{{\left(v,w,\frac{q}{as},q;q\right)}_k}\sum _{n=0}^{\infty }\frac{q^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}{(r{q}^k,f{q}^k,g{q}^k;q)}_n{\left(ut\right)}^n}{{(v{q}^k,w{q}^k,q;q)}_n}.\end{array}$$

(40)

Finally, by combining the above observations (38) to (40), we arrive at the assertion (34) of Theorem 1. This evidently completes the proof of Theorem 1. □

We remark that, when $g=w=0$, Theorem 1 reduces to the concluding result of Li and Tan [24].

Corollary 1.

It is asserted that

$$\begin{array}{c}\hfill \mathbb{T}(r,f,g,v,w,u{D}_s)\left\{\frac{1}{{(xs;q)}_{\infty }}\right\}=\frac{1}{{(xs;q)}_{\infty }}{}_3{\Phi }_2\left[\begin{array}{c}\begin{array}{c}\hfill r,f,g;\\ \\ \hfill v,w;\end{array}\end{array}q;xu\right]\end{array}$$

(41)

and

$$\begin{array}{c}\hfill \mathbb{E}(r,f,g,v,w,-u{\theta }_s)\left\{{(xs;q)}_{\infty }\right\}={(xs;q)}_{\infty }{}_3{\Phi }_3\left[\begin{array}{c}\begin{array}{c}\hfill r,f,g;\\ \\ \hfill v,w,0;\end{array}\end{array}q;xu\right],\end{array}$$

(42)

provided that $max\left\{\left|xs\right|,\left|xu\right|\right\}<1$.

The goal in this paper is to give potentially useful generalizations of a number q-series and q-integral identities such as the q-binomial theorem or the q-Gauss sum, the q-Chu-Vandermonde summation formula and the Andrews-Askey integral.

Our paper is organized as follows. In Section 2, we give two formal generalizations of the q-binomial theorem or the q-Gauss sum by applying the q-difference equations. In Section 3, we derive a set of two extensions q-Chu-Vandermonde summation formulas by making use of the q-difference equations. Next, in Section 4, we derive two new generalizations of the Andrews-Askey integral by means of the q-difference equations. Finally, in our last section (Section 5), we present a number of concluding remarks and observations concerning the various results which we have considered in this investigation.

2. A Set of Formal Generalizations of the q-Binomial Theorem

We begin this section by recalling the following q-binomial theorem (see, for example, [1,2,3,14]):

$$\sum _{n=0}^{\infty }\frac{{(a;q)}_n{x}^n}{{(q;q)}_n}=\frac{{(ax;q)}_{\infty }}{{(x;q)}_{\infty }}\left(\right|x|<1).$$

(43)

In Theorem 2 below, we give two generalizations of the q-binomial theorem (43) by applying the q-difference equations.

Theorem 2.

Each of the following assertions holds true:

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }\frac{{(a;q)}_n{a}^{-n}}{{(q;q)}_n}\sum _{k=0}^{\infty }\frac{{(q^{-n},ax;q)}_k{q}^k}{{(q;q)}_k}\sum _{j,i\geqq 0}\frac{{(r,f,g;q)}_{j+i}{u}^{j+i}}{{(q;q)}_i{(v,w;q)}_{j+i}}\frac{{\left(\frac{c}{b},ax{q}^k;q\right)}_j}{{(cx,q;q)}_j}{\left(a{q}^k\right)}^i{b}^j\hfill \\ \hfill & =\frac{{(ax;q)}_{\infty }}{{(x;q)}_{\infty }}\sum _{j,i\geqq 0}\frac{{(r,f,g;q)}_{j+i}{u}^{j+i}}{{(q;q)}_i{(v,w;q)}_{j+i}}\frac{{\left(\frac{c}{b},x;q\right)}_j}{{(cx,q;q)}_j}{b}^j\left(\right|x|<1)\hfill \end{array}$$

(44)

and

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }\frac{{(a;q)}_n{a}^{-n}}{{(q;q)}_n}\sum _{k=0}^{\infty }\frac{{(q^{-n},ax;q)}_k{q}^k}{{(q;q)}_k}\sum _{i,j\geqq 0}\frac{{(-1)}^{j+i}{q}^{\left({}_2^i\right)}{(r,f,g;q)}_{j+i}}{{(q;q)}_i{(v,w;q)}_{j+i}}\hfill \\ \hfill & ·\frac{{\left(\frac{b{q}^{1-k}}{a},\frac{q}{cx};q\right)}_j}{{\left(\frac{q^{1-k}}{ax},q;q\right)}_j}{\left(uc\right)}^{j+i}\hfill \\ \hfill & =\frac{{(ax;q)}_{\infty }}{{(x;q)}_{\infty }}\sum _{i,j=0}^{\infty }\frac{{(-1)}^{j+i}{q}^{\left({}_2^i\right)}{(r,f,g;q)}_{j+i}}{{(q;q)}_i{(v,w;q)}_{j+i}}\frac{{\left(\frac{q}{bx};q\right)}_j}{{\left(\frac{q}{x},q;q\right)}_j}{\left(bu\right)}^{j+i},\hfill \end{array}$$

(45)

provided that both sides of (44) and (45) exist.

Remark 1.

For $u=0$ and by using the fact that

$$\sum _{k=0}^{\infty }\frac{{(q^{-n},ax;q)}_k}{{(q;q)}_k}{q}^k={}_2{\Phi }_1\left[\begin{array}{c}\begin{array}{c}\hfill q^{-n},ax;\\ \\ \hfill 0;\end{array}\end{array}q;q\right]={\left(ax\right)}^n,$$

the assertions (44) and (45) reduce to (43).

In our proof of Theorem 2, we shall need Theorem 3 and Corollary 2 below.

Theorem 3.

Each of the following assertions holds true:

$$\begin{array}{cc}\hfill & \mathbb{T}(r,f,g,v,w,u{D}_x)\left\{\frac{p_n\left(x,\frac{y}{a}\right){(cx;q)}_{\infty }}{{(ax,bx;q)}_{\infty }}\right\}\hfill \\ \hfill & =\frac{{(y;q)}_n}{a^n}\frac{{(cx;q)}_{\infty }}{{(ax,bx;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{(q^{-n},ax;q)}_k{q}^k}{{(y,q;q)}_k}\hfill \\ \hfill & ·\sum _{j,i\geqq 0}\frac{{(r,f,g;q)}_{j+i}{u}^{j+i}}{{(q;q)}_i{(v,w;q)}_{j+i}}\frac{{\left(\frac{c}{b},ax{q}^k;q\right)}_j}{{(cx,q;q)}_j}{\left(a{q}^k\right)}^i{b}^j\hfill \end{array}$$

(46)

and

$$\begin{array}{cc}\hfill & \mathbb{E}(r,f,g,v,w,u{\theta }_x)\left\{\frac{p_n\left(x,\frac{y}{a}\right){(bx,cx;q)}_{\infty }}{{(ax;q)}_{\infty }}\right\}\hfill \\ \hfill & =\frac{{(y;q)}_n}{a^n}\frac{{(bx,cx;q)}_{\infty }}{{(ax;q)}_{\infty }}\sum _{i,j\geqq 0}\frac{{(-1)}^{j+i}{q}^{\left({}_2^i\right)}{(r,f,g;q)}_{j+i}}{{(q;q)}_i{(v,w;q)}_{j+i}}\hfill \\ \hfill & ·\frac{{\left(\frac{b{q}^{1-k}}{a},\frac{q}{cx};q\right)}_j}{{\left(\frac{q^{1-k}}{ax},q;q\right)}_j}{\left(uc\right)}^{j+i},\hfill \end{array}$$

(47)

provided that $max\left\{\left|ax\right|,\left|bx\right|,\left|cx\right|\right\}<1$.

Corollary 2.

Each of the following assertions holds true:

$$\begin{array}{cc}\hfill & \mathbb{T}(r,f,g,v,w,u{D}_x)\left\{\frac{x^n{(cx;q)}_{\infty }}{{(ax,bx;q)}_{\infty }}\right\}\hfill \\ \hfill & =\frac{1}{a^n}\frac{{(cx;q)}_{\infty }}{{(ax,bx;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{(q^{-n},ax;q)}_k{q}^k}{{(q;q)}_k}\hfill \\ \hfill & ·\sum _{j,i\geqq 0}\frac{{(r,f,g;q)}_{j+i}{u}^{j+i}}{{(q;q)}_i{(v,w;q)}_{j+i}}\frac{{\left(\frac{c}{b},ax{q}^k;\right)}_j}{{(cx,q;q)}_j}{\left(a{q}^k\right)}^i{b}^j\hfill \end{array}$$

(48)

and

$$\begin{array}{cc}\hfill & \mathbb{E}(r,f,g,v,w,u{\theta }_x)\left\{\frac{x^n{(cx,bx;q)}_{\infty }}{{(ax;q)}_{\infty }}\right\}\hfill \\ \hfill & =\frac{1}{a^n}\frac{{(bx,cx;q)}_{\infty }}{{(ax;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{(q^{-n},ax;q)}_k{q}^k}{{(q;q)}_k}\sum _{i,j\geqq 0}\frac{{(-1)}^{j+i}{q}^{\left({}_2^i\right)}{(r,f,g;q)}_{j+i}}{{(q;q)}_i{(v,w;q)}_{j+i}}\hfill \\ \hfill & ·\frac{{\left(\frac{b{q}^{1-k}}{a},\frac{q}{cx};q\right)}_j}{{\left(\frac{q^{1-k}}{ax},q;q\right)}_j}{\left(cu\right)}^{j+i},\hfill \end{array}$$

(49)

provided that $max\left\{\left|ax\right|,\left|bx\right|,\left|cx\right|,\left|cu\right|\right\}<1$.

Remark 2.

For $y=0,$ the assertions (46) and (47) reduce to (48) and (49), respectively.

Proof of Theorem 3.

Upon first setting $x\to ax$ in (55) and then multiplying both sides of the resulting equation by $\frac{{(cx;q)}_{\infty }}{{(bx;q)}_{\infty }},$ we get

$$\sum _{n=0}^{\infty }\frac{{(q^{-n};q)}_k{q}^k}{{(y,q;q)}_k}\frac{{(cx;q)}_{\infty }}{{(ax{q}^k,bx;q)}_{\infty }}=\frac{{\left(ax\right)}^n{\left(\frac{y}{ax};q\right)}_n{(cx;q)}_{\infty }}{{(y;q)}_n{(ax,bx;q)}_{\infty }}.$$

(50)

Now, by applying the operator $\mathbb{T}(r,f,g,v,w,u{D}_x)$ to both sides of (50), it is easy to see that

$$\begin{array}{cc}\hfill & \sum _{k=0}^{\infty }\frac{{(q^{-n};q)}_k{q}^k}{{(y,q;q)}_k}\mathbb{T}(r,f,g,v,w,u{D}_x)\left\{\frac{{(cx;q)}_{\infty }}{{(ax{q}^k,bx;q)}_{\infty }}\right\}\hfill \\ \hfill & =\frac{a^n}{{(y;q)}_n}\mathbb{T}(r,f,g,v,w,u{D}_x)\left\{\frac{x^n{\left(\frac{y}{ax};q\right)}_n{(cx;q)}_{\infty }}{{(ax,bx;q)}_{\infty }}\right\}\hfill \\ \hfill & =\frac{a^n}{{(y;q)}_n}\mathbb{T}(r,f,g,v,w,u{D}_x)\left\{\frac{p_n\left(x,\frac{y}{a}\right){(cx;q)}_{\infty }}{{(ax,bx;q)}_{\infty }}\right\}.\hfill \end{array}$$

(51)

The proof of the first assertion (46) of Theorem 3 is completed by using the relation (33) in the left-hand side of (51).

The proof of the second assertion (47) of Theorem 3 is much akin to that of the first assertion (46). The details involved are, therefore, being omitted here. □

Proof of Theorem 2.

Multiplying both sides of (43) by $\frac{{(cx;q)}_{\infty }}{{(bx;q)}_{\infty }}$, we find that

$$\sum _{n=0}^{\infty }\frac{{(a;q)}_n}{{(q;q)}_n}\frac{x^n{(cx;q)}_{\infty }}{{(ax,bx;q)}_{\infty }}=\frac{{(cx;q)}_{\infty }}{{(bx,x;q)}_{\infty }}.$$

(52)

Equation (44) can be written equivalently as follows:

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }\frac{{(a;q)}_n}{{(q;q)}_n}·\frac{a^{-n}{(cx;q)}_{\infty }}{{(bx,ax;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{(q^{-n},ax;q)}_k{q}^k}{{(q;q)}_k}\hfill \\ \hfill & ·\sum _{j,i\geqq 0}\frac{{(r,f,g;q)}_{j+i}{u}^{j+i}}{{(q;q)}_i{(v,w;q)}_{j+i}}\frac{{\left(\frac{c}{b},ax{q}^k;q\right)}_j}{{(cx,q;q)}_j}{\left(a{q}^k\right)}^i{b}^j\hfill \\ \hfill & =\frac{{(cx;q)}_{\infty }}{{(bx,x;q)}_{\infty }}\sum _{j,i=0}\frac{{(r,f,g;q)}_{j+i}{u}^{j+i}}{{(q;q)}_i{(v,w;q)}_{j+i}}·\frac{{\left(\frac{c}{b},x;q\right)}_j}{{(cx,q;q)}_j}{b}^j.\hfill \end{array}$$

(53)

If we use $F(r,f,g,v,w,a,u)$ to denote the right-hand side of (53), it is easy to verify that $F(r,f,g,v,w,a,u)$ satisfies (23). By applying (24), we thus find that

$$\begin{array}{cc}\hfill F(r,f,g,v,w,x,u)& =\mathbb{T}(r,f,g,v,w,u{D}_x)\left\{F(r,f,g,v,w,x,0)\right\}\hfill \\ \hfill & =\mathbb{T}(r,f,g,v,w,u{D}_x)\left\{\frac{{(cx;q)}_{\infty }}{{(bx,x;q)}_{\infty }}\right\}\hfill \\ \hfill & =\mathbb{T}(r,f,g,v,w,u{D}_x)\left\{\sum _{n=0}^{\infty }\frac{{(a;q)}_n}{{(q;q)}_n}\frac{x^n{(cx;q)}_{\infty }}{{(ax,bx;q)}_{\infty }}\right\}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\frac{{(a;q)}_n}{{(q;q)}_n}\mathbb{T}(r,f,g,v,w,u{D}_x)\left\{\frac{x^n{(cx;q)}_{\infty }}{{(ax,bx;q)}_{\infty }}\right\}.\hfill \end{array}$$

(54)

The proof of the first assertion (44) of Theorem 2 can now be completed by making use of the relation (48).

The proof of the second assertion (45) of Theorem 2 is much akin to that of the first assertion (44). The details involved are, therefore, being omitted here. □

3. Two Generalizations of the q-Chu-Vandermonde Summation Formula

The q-Chu-Vandermonde summation formula is recalled here as follows (see, for example, [1,31]):

$${}_2{\Phi }_1\left[\begin{array}{c}\begin{array}{c}\hfill q^{-n},x;\\ \\ \hfill y;\end{array}\end{array}q;q\right]=\frac{{\left(\frac{y}{x};q\right)}_n}{{(y;q)}_n}{x}^n\left(n\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}\right).$$

(55)

In this section, we give two generalizations of the q-Chu-Vandermonde summation Formula (55) by applying q-difference equations.

Theorem 4.

The following assertion holds true for $y\ne 0$:

$$\begin{array}{cc}\hfill & \sum _{k=0}^n\frac{{(q^{-n},x;q)}_k{q}^k}{{(q,y;q)}_k}{}_3{\Phi }_2\left[\begin{array}{c}\begin{array}{c}\hfill r,f,g;\\ \\ \hfill v,w;\end{array}\end{array}q;u{q}^k\right]\hfill \\ \hfill & =\frac{x^n{\left(\frac{y}{x};q\right)}_n}{{(y;q)}_n}\sum _{k,j\geqq 0}\frac{{(r,f,g;q)}_{k+j}}{{(q;q)}_j{(v,w;q)}_{k+j}}\frac{{\left(\frac{q^{1-n}}{y},\frac{qx}{y};q\right)}_k}{{\left(\frac{x{q}^{1-n}}{y},q;q\right)}_k}{u}^{k+j}{\left(\frac{q}{y}\right)}^j.\hfill \end{array}$$

(56)

We next derive another generalization of the q-Chu-Vandermonde summation Formula (55) as follows.

Theorem 5.

For $m\in {\mathbb{N}}_0$ and $y\ne 0$, it is asserted that

$$\begin{array}{c}\hfill {}_2{\Phi }_1\left[\begin{array}{c}\begin{array}{c}\hfill q^{-n},x;\\ \\ \hfill y;\end{array}\end{array}q;q^{1+m}\right]=\frac{x^n{\left(\frac{y}{x};q\right)}_n}{{(y;q)}_n}\sum _{j=0}^m{\left[\begin{array}{c}m\\ j\end{array}\right]}_q\frac{{\left(\frac{q^{1-n}}{y},\frac{qx}{y};q\right)}_{m-j}}{{\left(\frac{x{q}^{1-n}}{y};q\right)}_{m-j}}{\left(\frac{q}{y}\right)}^j.\end{array}$$

(57)

Remark 3.

For $u=0$ or $m=0,$ the assertion (56) or (57) reduces to the q-Chu-Vandermonde summation Formula (55). Furthermore, if we first set $i+j=m$ and then extract the coefficients of ${\displaystyle \frac{{(r,f,g;q)}_m}{{(v,w;q)}_m}{u}^m}$ from the two members of the assertion (56) of Theorem 4, we obtain the transformation formula (57), which leads us to the q-Chu-Vandermonde summation Formula (55) when $m=0$. Also, upon putting $n=0,$ the assertion (57) reduces to the following identity:

$$\begin{array}{c}\hfill \sum _{j=0}^m{\left[\begin{array}{c}m\\ j\end{array}\right]}_q{\left(\frac{q}{y};q\right)}_{m-j}{\left(\frac{q}{y}\right)}^j=1(y\ne 0).\end{array}$$

(58)

Proof of Theorem 4.

We first write (55) in the following form:

$$\sum _{k=0}^n\frac{{(q^{-n};q)}_k{q}^k}{{(y,q;q)}_k}\frac{1}{{(x{q}^k;q)}_{\infty }}=\frac{{(-1)}^n{y}^n{q}^{\left({}_2^n\right)}}{{(y;q)}_n}\frac{{\left(\frac{x{q}^{1-n}}{y};q\right)}_{\infty }}{{\left(x,\frac{qx}{y};q\right)}_{\infty }}.$$

(59)

Equation (56) can be written equivalently as follows:

$$\begin{array}{cc}\hfill & \sum _{k=0}^{\infty }\frac{{(q^{-n};q)}_k{q}^k}{{(q,c;q)}_k}·\frac{1}{{(x{q}^k;q)}_{\infty }}{}_3{\Phi }_2\left[\begin{array}{c}\begin{array}{c}\hfill r,f,g;\\ \\ \hfill v,w;\end{array}\end{array}q;u{q}^k\right]\hfill \\ \hfill & =\frac{{(-1)}^n{y}^n{q}^{\left({}_2^n\right)}}{{(y;q)}_n}\frac{{\left(\frac{x{q}^{1-n}}{y};q\right)}_{\infty }}{{\left(x,\frac{qx}{y};q\right)}_{\infty }}·\sum _{i,j\geqq 0}\frac{{(r,f,g;q)}_{j+i}{u}^{j+i}}{{(q;q)}_i{(v,w;q)}_{j+i}}\frac{{\left(\frac{q^{1-n}}{y},\frac{qx}{y};q\right)}_j}{{\left(\frac{x{q}^{1-n}}{y},q;q\right)}_j}{\left(\frac{q}{y}\right)}^i.\hfill \end{array}$$

(60)

If we use $G(r,f,g,v,w,x,u)$ to denote the right-hand side of (60), it is easy to observe that $G(r,f,g,v,w,x,u)$ satisfies (23). By using (24), we obtain

$$\begin{array}{cc}\hfill G(r,f,g,v,w,x,u)& =\mathbb{T}(r,f,g,v,w,u{D}_x)\left\{G(r,f,g,v,w,x,0)\right\}\hfill \\ \hfill & =\mathbb{T}(r,f,g,v,w,u{D}_x)\left\{\frac{{(-1)}^n{y}^n{q}^{\left({}_2^n\right)}}{{(y;q)}_n}\frac{{\left(\frac{x{q}^{1-n}}{y};q\right)}_{\infty }}{{\left(x,\frac{qx}{y};q\right)}_{\infty }}\right\}\hfill \\ \hfill & =\mathbb{T}(r,f,g,v,w,u{D}_x)\left\{\sum _{k=0}^{\infty }\frac{{(q^{-n};q)}_k{q}^k}{{(y,q;q)}_k}\frac{1}{{(x{q}^k;q)}_{\infty }}\right\}\hfill \\ \hfill & =\sum _{k=0}^n\frac{{(q^{-n};q)}_k{q}^k}{{(y,q;q)}_k}\mathbb{T}(r,f,g,v,w,u{D}_x)\left\{\frac{1}{{(x{q}^k;q)}_{\infty }}\right\}.\hfill \end{array}$$

(61)

Finally, by using the fact that

$$\mathbb{T}(r,f,g,v,w,u{D}_x)\left\{\frac{1}{{(x{q}^k;q)}_{\infty }}\right\}=\frac{1}{{(x{q}^k;q)}_{\infty }}{}_3{\Phi }_2\left[\begin{array}{c}\begin{array}{c}\hfill r,f,g;\\ \\ \hfill v,w;\end{array}\end{array}q;u{q}^k\right],$$

(62)

and after some simplification involving $\frac{1}{{(x;q)}_{\infty }}$, we get the left-hand side of (56). □

4. New Generalizations of the Andrews-Askey Integral

The following famous formula is known as the Andrews-Askey integral (see, for details, [32]). It was derived from Ramanujan’s celebrated ${}_1{\Psi }_1$-summation formula.

Proposition 2.

(see ([32], Equation (2.1))). For $max\left\{\left|ac\right|,\left|ad\right|,\left|bc\right|,\left|bd\right|\right\}<1,$ it is asserted that

$$\begin{array}{c}\hfill {\int }_c^d\frac{{\left(\frac{qt}{c},\frac{qt}{d};q\right)}_{\infty }}{{(at,bt;q)}_{\infty }}{\mathrm{d}}_{q}t=\frac{d(1-q){\left(q,\frac{dq}{c},\frac{c}{d},abcd;q\right)}_{\infty }}{{(ac,ad,bc,bd;q)}_{\infty }}.\end{array}$$

(63)

The Andrews-Askey integral (63) is indeed an important formula in the theory of q-series (see [16]).

Recently, Cao [33] gave the following two generalizations of the Andrews-Askey integral (63) by the method based upon q-difference equations.

Proposition 3.

(see ([33], Theorems 14 and 15)) For $N\in \mathbb{N}$ and $r=q^{-N},$ suppose that

$$max\left\{\left|ac\right|,\left|ad\right|,\left|bc\right|,\left|bd\right|,\left|\frac{qwr}{v}\right|,\left|\frac{q}{v}\right|\right\}<1.$$

Then

$$\begin{array}{cc}\hfill & {\int }_c^d\frac{{\left(\frac{qt}{c},\frac{qt}{d};q\right)}_{\infty }}{{(at,bt;q)}_{\infty }}{}_4{\Phi }_2\left[\begin{array}{c}\begin{array}{c}\hfill r,w,\frac{c}{t},abcd;\\ \\ \hfill ac,\frac{qwr}{v};\end{array}\end{array}q;\frac{qt}{vbcd}\right]{\mathrm{d}}_{q}t\hfill \\ \hfill & =\frac{d(1-q){\left(q,\frac{dq}{c},\frac{c}{d},abcd,\frac{qw}{v},\frac{qr}{v};q\right)}_{\infty }}{{\left(ac,ad,bc,bd,\frac{qwr}{v},\frac{q}{v};q\right)}_{\infty }}{}_2{\Phi }_1\left[\begin{array}{c}\begin{array}{c}\hfill w,r;\\ \\ \hfill v;\end{array}\end{array}q;\frac{q}{bc}\right].\hfill \end{array}$$

(64)

Furthermore, for $N\in \mathbb{N}$ and $r=q^{-N},$ suppose that

$$max\left\{\left|ac\right|,\left|ad\right|,\left|bc\right|,\left|bd\right|,\left|\frac{v}{w}\right|,\left|\frac{v}{r}\right|\right\}<1.$$

Then

$$\begin{array}{cc}\hfill & {\int }_c^d\frac{{\left(\frac{qt}{c},\frac{qt}{d};q\right)}_{\infty }}{{(at,bt;q)}_{\infty }}{}_4{\Phi }_2\left[\begin{array}{c}\begin{array}{c}\hfill r,w,\frac{c}{t},\frac{q}{ad};\\ \\ \hfill \frac{q}{at},\frac{qrw}{v};\end{array}\end{array}q;q\right]d_{q}t\hfill \\ \hfill & =\frac{d(1-q){\left(q,\frac{dq}{c},\frac{c}{d},abcd,\frac{v}{wr},v;q\right)}_{\infty }}{{\left(ac,ad,bc,bd,\frac{v}{w},\frac{v}{r};q\right)}_{\infty }}{}_2{\Phi }_1\left[\begin{array}{c}\begin{array}{c}\hfill w,r;\\ \\ \hfill v;\end{array}\end{array}q;\frac{vbc}{wr}\right].\hfill \end{array}$$

(65)

In this section, we give the following two generalizations of the Andrews-Askey integral (63) by using the method of q-difference equations.

Theorem 6.

For $M\in \mathbb{N}$ and $r=q^{-M},$ suppose that

$$max\left\{\left|ac\right|,\left|ad\right|,\left|bc\right|,\left|bd\right|,\left|\frac{q}{bc}\right|\right\}<1.$$

Then

$$\begin{array}{cc}\hfill & {\int }_c^d\frac{{\left(\frac{qt}{c},\frac{qt}{d};q\right)}_{\infty }}{{(at,bt;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{\left(r,f,g,\frac{c}{t},abcd;q\right)}_k{\left(\frac{qt}{bcd}\right)}^k}{{(v,w,ac,q;q)}_k}{}_3{\Phi }_2\left[\begin{array}{c}\begin{array}{c}\hfill r{q}^k,f{q}^k,g{q}^k;\\ \\ \hfill v{q}^k,w{q}^k;\end{array}\end{array}q;q\right]{\mathrm{d}}_{q}t\hfill \\ \hfill & =\frac{d(1-q){\left(q,\frac{dq}{c},\frac{c}{d},abcd;q\right)}_{\infty }}{{(ac,ad,bc,bd;q)}_{\infty }}{}_3{\Phi }_2\left[\begin{array}{c}\begin{array}{c}\hfill r,f,g;\\ \\ \hfill v,w;\end{array}\end{array}q;\frac{q}{bc}\right].\hfill \end{array}$$

(66)

Theorem 7.

For $M\in \mathbb{N}$ and $r=q^{-M},$ suppose that $max\left\{\left|ac\right|,\left|ad\right|,\left|bc\right|,\left|bd\right|\right\}<1$. Then

$$\begin{array}{cc}\hfill & {\int }_c^d\frac{{\left(\frac{qt}{c},\frac{qt}{d};q\right)}_{\infty }}{{(at,bt;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{\left(r,f,g,\frac{c}{t},\frac{q}{ad};q\right)}_k{\left(\frac{vw}{rfg}\right)}^k}{{\left(v,w,\frac{q}{at},q;q\right)}_k}{}_3{\Phi }_3\left[\begin{array}{c}\begin{array}{c}\hfill r{q}^k,f{q}^k,g{q}^k;\\ \\ \hfill v{q}^k,w{q}^k,0;\end{array}\end{array}q;-\frac{vw}{rfg}\right]{\mathrm{d}}_{q}t\hfill \\ \hfill & =\frac{d(1-q){\left(q,\frac{dq}{c},\frac{c}{d},abcd;q\right)}_{\infty }}{{(ac,ad,bc,bd;q)}_{\infty }}{}_3{\Phi }_3\left[\begin{array}{c}\begin{array}{c}\hfill r,f,g;\\ \\ \hfill v,w,0;\end{array}\end{array}q;\frac{vwbc}{rfg}\right].\hfill \end{array}$$

(67)

Remark 4.

For $r=1,$ both (66) and (67) reduce to (63). Moreover, for $r=q^{-N},$$g=w=0$ and $u=\frac{q}{bcd},$ the assertion (66) of Theorem 6 reduces to (64). For $r=q^{-N},$$g=w=0$ and $u=\frac{v}{rfbcd},$ the assertion (66) of Theorem 7 reduces to (67).

Proof of Theorems 6 and 7.

Equation (66) can be written equivalently as follows:

$$\begin{array}{cc}\hfill & {\int }_c^d\frac{{\left(\frac{qt}{c},\frac{qt}{d};q\right)}_{\infty }}{{(bt;q)}_{\infty }}·\frac{{(ac;q)}_{\infty }}{{(at,abcd;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{\left(r,f,g,\frac{c}{t},abcd;q\right)}_k{\left(\frac{qt}{bcd}\right)}^k}{{(v,w,ac,q;q)}_k}\hfill \\ \hfill & ·{}_3{\Phi }_2\left[\begin{array}{c}\begin{array}{c}\hfill r{q}^k,f{q}^k,g{q}^k;\\ \\ \hfill v{q}^k,w{q}^k;\end{array}\end{array}q;q\right]{\mathrm{d}}_{q}t\hfill \\ \hfill & =\frac{d(1-q){\left(q,\frac{dq}{c},\frac{c}{d};q\right)}_{\infty }}{{(bc,bd;q)}_{\infty }}·\frac{1}{{(ad;q)}_{\infty }}{}_3{\Phi }_2\left[\begin{array}{c}\begin{array}{c}\hfill r,f,g;\\ \\ \hfill v,w;\end{array}\end{array}q;\frac{q}{bc}\right].\hfill \end{array}$$

(68)

If we use $H(r,f,g,v,w,a,u)$ to denote the right-hand side of (68), it is easy to see that $H(r,f,g,v,w,a,u)$ satisfies (23) with $u=\frac{q}{bcd}$. By making use of (24), we thus find that

$$\begin{array}{cc}\hfill H(r,f,g,v,w,a,u)& =\mathbb{T}(r,f,g,v,w,u{D}_a)\left\{H(r,f,g,v,w,a,0)\right\}\hfill \\ \hfill & =\mathbb{T}\left(r,f,g,v,w,\frac{q}{bcd}{D}_a\right)\left\{H(1,f,g,v,w,a,u)\right\}\hfill \\ \hfill & =\mathbb{T}\left(r,f,g,v,w,\frac{q}{bcd}{D}_a\right)\left\{\frac{d(1-q){\left(q,\frac{dq}{c},\frac{c}{d};q\right)}_{\infty }}{{(bc,bd;q)}_{\infty }}·\frac{1}{{(ad;q)}_{\infty }}\right\}\hfill \\ \hfill & =\mathbb{T}\left(r,f,g,v,w,\frac{q}{bcd}{D}_a\right)\left\{{\int }_c^d\frac{{\left(\frac{qt}{c},\frac{qt}{d};q\right)}_{\infty }}{{(bt;q)}_{\infty }}·\frac{{(ac;q)}_{\infty }}{{(at,abcd;q)}_{\infty }}{\mathrm{d}}_{q}t\right\}\hfill \\ \hfill & ={\int }_c^d\frac{{\left(\frac{qt}{c},\frac{qt}{d};q\right)}_{\infty }}{{(bt;q)}_{\infty }}·\mathbb{T}\left(r,f,g,v,w,\frac{q}{bcd}{D}_a\right)\left\{\frac{{(ac;q)}_{\infty }}{{(at,abcd;q)}_{\infty }}\right\}{\mathrm{d}}_{q}t.\hfill \end{array}$$

Now, by applying the fact that

$$\begin{array}{cc}\hfill & \mathbb{T}\left(r,f,g,v,w,\frac{q}{bcd}{D}_a\right)\left\{\frac{{(ac;q)}_{\infty }}{{(at,abcd;q)}_{\infty }}\right\}\hfill \\ \hfill & =\frac{{(ac;q)}_{\infty }}{{(at,abcd;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{\left(r,f,g,\frac{c}{t},abcd;q\right)}_k{\left(\frac{qt}{bcd}\right)}^k}{{(v,w,ac,q;q)}_k}{}_3{\Phi }_2\left[\begin{array}{c}\begin{array}{c}\hfill r{q}^k,f{q}^k,g{q}^k;\\ \\ \hfill v{q}^k,w{q}^k;\end{array}\end{array}q;q\right],\hfill \end{array}$$

we get the left-hand side of (66).

Similarly, Equation (67) can be written equivalently as follows:

$$\begin{array}{cc}\hfill & {\int }_c^d\frac{{\left(\frac{qt}{c},\frac{qt}{d};q\right)}_{\infty }}{{(bt;q)}_{\infty }}·\frac{{\left(ac;q\right)}_{\infty }}{{(at,abcd;q)}_{\infty }}\sum _{k=0}^{\infty }\frac{{\left(r,f,g,\frac{c}{t},\frac{q}{ad};q\right)}_k{\left(\frac{vw}{rfg}\right)}^k}{{\left(v,w,\frac{q}{at},q;q\right)}_k}\hfill \\ \hfill & ·{}_3{\Phi }_3\left[\begin{array}{c}\begin{array}{c}\hfill r{q}^k,f{q}^k,g{q}^k;\\ \\ \hfill v{q}^k,w{q}^k,0;\end{array}\end{array}q;-\frac{vw}{rfg}\right]{\mathrm{d}}_{q}t\hfill \\ \hfill & =\frac{d(1-q){\left(q,\frac{dq}{c},\frac{c}{d};q\right)}_{\infty }}{{(bc,bd;q)}_{\infty }}·\frac{1}{{(ad;q)}_{\infty }}{}_3{\Phi }_3\left[\begin{array}{c}\begin{array}{c}\hfill r,f,g;\\ \\ \hfill v,w,0;\end{array}\end{array}q;\frac{vwbc}{rfg}\right].\hfill \end{array}$$

(69)

The proof of the assertion (67) of Theorem 7 is much akin to that of the assertion (66) of Theorem 6 by using Proposition 1 (II). The analogous details involved are, therefore, being omitted here.

The proofs of Theorems 6 and 7 are thus completed. □

5. Concluding Remarks and Observations

In our present investigation, we have introduced a set of two q-operators $\mathbb{T}(a,b,c,d,e,y{D}_x)$ and $\mathbb{E}(a,b,c,d,e,y{\theta }_x)$ with to applying them to derive two potentially useful generalizations of the q-binomial theorem, two extensions of the q-Chu-Vandermonde summation formula and two new generalizations of the Andrews-Askey integral by means of the q-difference equations. We have also briefly described relevant connections of various special cases and consequences of our main results with several known results.

It is believed that the q-series and q-integral identities, which we have presented in this paper, as well as the various related recent works cited here, will provide encouragement and motivation for further researches on the topics that are dealt with and investigated in this paper.

Just as we mentioned in Section 1 above, basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas [see also ([3], pp. 350–351)]. In particular, the celebrated Chu-Vandermonde summation theorem and its known q-extensions, which we have considered in this paper, have already been demonstrated to be useful (see, for details, [1,14,31]). The new q-Chu-Vandermone summation theorems, which we have presented in this paper, are believed to be useful as well.

In conclusion, we find it to be worthwhile to remark that some potential further applications of the methodology and findings, which we have presented here by means of the q-analysis and the q-calculus, can be found in the study of the zeta and q-zeta functions as well as their related functions of Analytic Number Theory (see, for example, [34,35]; see also [14]) and also in the study of analytic and univalent functions of Geometric Function Theory via number-theoretic objects (see, for example, [36]).