Applications of q-Real Numbers to Triple q-Hypergeometric Functions and q-Horn Functions

Abstract

The purpose of this article is to study how q-real numbers can be used for computations of convergence regions, q-integral representations of certain multiple triple q-Lauricella functions. The corresponding q-difference equations are also given without proof. In the process, we slightly improve Exton’s original formulas. We also survey the current attempts to generalize the above functions to triple and quadruple hypergeometric functions. Finally, we compute some q-analogues of transformation formulas for Horn functions.

1. Introduction

This is part of a series of papers about multiple q-hypergeometric functions. Many attempts have been made to generalize the Appell, Lauricella and Lauricella triple functions by Exton [1,2], by Srivastava [3,4] (to four variables), by Qurechi et al. [5] (to four variables), etc. The most successful generalization was by Karlsson [6], who introduced ‘symmetric’ intermediate Lauricella functions. It seems that ‘more general forms’ of multiple hypergeometric functions, which lack symmetry, do not give concise, short formulas. In this paper, we shall q-deform some of Exton’s contributions to multiple hypergeometric functions. The paper is organized as follows: The first q-calculus definitions are given in Section 2. Some of the q-real numbers are briefly introduced in Section 3. The confluent triple q-Lauricella functions are treated in Section 4. For the increased convergence regions in the confluent case, see [7]. Furthermore, q-integral representations with q-real numbers and canonical q-difference equations are given.

Summation formulas for q-Appell functions were given in our book [8]. In the same vein, a summation formula for the second q-Appell function is proved in Section 5. Several so-called q-Horn functions were introduced in our work [7]. In the same spirit, transformation formulas for some other q-Horn functions are proved in Section 6.

2. First $q$-Calculus Definitions

We now repeat some notation from [8]. Let $\delta >0$ be an arbitrary small number. We will always use the following branch of the logarithm: $-\pi +\delta <\mathrm{Im}(logq)\le \pi +\delta$. This defines a simply connected space in the complex plane.

The power function is defined by

$$q^a\equiv e^{alog\left(q\right)}.$$

(1)

A q-analogue of a complex number is also a complex number. The following notation is often used when we have long exponents.

$$\mathrm{QE}\left(x\right)\equiv q^x.$$

(2)

Definition 1.

The q-analogue of a complex number a is defined as follows:

$${\left\{a\right\}}_q\equiv \frac{1-q^a}{1-q},q\in \mathbb{C}\setminus \{0,1\}.$$

(3)

The q-shifted factorial is defined by

$${\displaystyle {\langle a;q\rangle }_n\equiv \prod _{m=0}^{n-1}(1-q^{a+m}).}$$

(4)

The q-derivative is defined by

$${\displaystyle \left(D_q\phi \right)\left(x\right)\equiv \left\{\begin{array}{cc}{\displaystyle \frac{\phi \left(x\right)-\phi \left(qx\right)}{(1-q)x},}\hfill & \mathrm{when} q\in \mathbb{C}\setminus \left\{1\right\},x\ne 0;\hfill \\ {\displaystyle \frac{d\phi }{dx}\left(x\right),}\hfill & \mathrm{when} q=1;\hfill \\ {\displaystyle \frac{d\phi }{dx}\left(0\right),}\hfill & \mathrm{when} x=0.\hfill \end{array}\right.}$$

(5)

Definition 2.

The following operator will also be useful.

$${\theta }_{q,j}\equiv x_j{D}_{q,x_j}.$$

(6)

Definition 3

([8], p. 212). The multiplication operator $\mathbb{C}\left[\left[\overrightarrow{x}\right]\right]\to \mathbb{C}\left[\left[\overrightarrow{x}\right]\right]$, i.e., multiplication with $x_i,0\le i\le n$, is denoted by ${\mathbf{x}}_i$. We have skipped the I for the unit operator to the far right in all q-difference equations.

3. Survey of $\mathbf{q}$-Real Numbers

The q-real numbers give a convenient notation for q-additions in formal power series, in particular for q-exponential and q-trigonometric functions. There is a one-to-one correspondence between the convergence regions of the two q-Lauricella functions, ${\mathsf{\Phi }}_A^{\left(n\right)}$ and ${\mathsf{\Phi }}_C^{\left(n\right)}$ [9], and the existence of q-real numbers with n letters (or variables). There are three types of q-real numbers [10], ${\mathbb{R}}_{{\oplus }_q},{\mathbb{R}}_q$ and ${\mathbb{R}}_{{⊞}_q}$. We just give a brief survey of these three q-real number definitions.

Definition 4.

The q-binomial coefficients [8] are defined by

$${\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_q\equiv \frac{{\langle 1;q\rangle }_n}{{\langle 1;q\rangle }_k{\langle 1;q\rangle }_{n-k}},k=0,1,\dots ,n.$$

(7)

Theorem 1.

([8]). The q-binomial coefficient ${\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_q$ is a polynomial of degree $k(n-k)$ in q with integer coefficients, whose sum equals $\left(\genfrac{}{}{0pt}{}{n}{k}\right)$.

Definition 5

([8], p. 24). Let $a,b\in \mathbb{R}$. Then, the NWA q-addition is given by

$${\left(a{\oplus }_{q}b\right)}^n\equiv \sum _{k=0}^n{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_q{a}^k{b}^{n-k},n=0,1,2,\dots ,a{\oplus }_{q}b\in {\mathbb{R}}_{{\oplus }_q}.$$

(8)

In particular, ${\left(a{\oplus }_{q}b\right)}^0\equiv 1$. Furthermore, we put

$${\left(a{\ominus }_{q}b\right)}^n\equiv \sum _{k=0}^n{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_q{a}^k{(-b)}^{n-k},n=0,1,2,\dots .$$

(9)

Definition 6.

Let $a,b\in \mathbb{R}$. The Jackson–Hahn–Cigler q-addition (JHC) is the function

$$\begin{array}{cc}\hfill & {\left(a{⊞}_{q}b\right)}^n\equiv \sum _{k=0}^n{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_q{q}^{\left(\genfrac{}{}{0pt}{}{k}{2}\right)}{b}^k{a}^{n-k}=a^n{(-\frac{b}{a};q)}_n,n=0,1,2,\dots ,a{⊞}_{q}b\in {\mathbb{R}}_q.\hfill \end{array}$$

(10)

The JHC q-subtraction is defined analogously:

$${\left(a{\boxminus }_{q}b\right)}^n\equiv \sum _{k=0}^n{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_q{q}^{\left(\genfrac{}{}{0pt}{}{k}{2}\right)}{(-b)}^k{a}^{n-k},n=0,1,2,\dots$$

(11)

The JHC q-addition is neither commutative nor associative. For the commutative monoid ${\mathbb{R}}_{{\oplus }_q}$, we note the following definitions and formulas:

Theorem 2.

Assume that ∼ means equality on $\mathbb{R}\left[x\right]]$ ([8], p. 101). The q-addition (8) has the following properties, for $\alpha ,\beta ,\gamma \in {\mathbb{R}}_{{\oplus }_q}$:

Commutativity:

$$\alpha {\oplus }_q\beta \sim \beta {\oplus }_q\alpha .$$

(12)

Associativity

$$\left(\alpha {\oplus }_q\beta \right){\oplus }_q\gamma \sim \alpha {\oplus }_q\left(\beta {\oplus }_q\gamma \right).$$

(13)

Definition 7

([8]). The q-multinomial coefficient is defined by

$${\left(\genfrac{}{}{0pt}{}{n}{k_1,k_2,\dots ,k_m}\right)}_q\equiv \frac{{\langle 1;q\rangle }_n}{{\langle 1;q\rangle }_{k_1}{\langle 1;q\rangle }_{k_2}\cdots {\langle 1;q\rangle }_{k_m}},$$

(14)

where $k_1+k_2+\cdots +k_m=n$. If the number of $k_i$ is unspecified for $m=\infty$ in (14), we denote the q-multinomial coefficients by

$${\left(\genfrac{}{}{0pt}{}{n}{\overrightarrow{k}}\right)}_q,\sum _{i=1}^{\infty }{k}_i=n.$$

(15)

For $\overrightarrow{m}\in {\mathbb{N}}^n$, put

$$|\overrightarrow{m}|\equiv m_1+\dots +m_n.$$

The q-real number ${\mathbb{R}}_{{\oplus }_q}$, which appears inside the paranthesis of (16), is defined by

$${(a_1{\oplus }_q{a}_2{\oplus }_q\cdots {\oplus }_q{a}_n)}^k\equiv \sum _{|\overrightarrow{m}|=k}\prod _{l=1}^n{\left(a_l\right)}^{m_l}{\left(\genfrac{}{}{0pt}{}{k}{\overrightarrow{m}}\right)}_q.$$

(16)

The q-real number ${\mathbb{R}}_q$, which appears inside the parenthesis of (17), is defined by

$$\begin{array}{cc}& F\left(k\right)\equiv {(a_1{\oplus }_q{a}_2{\oplus }_q\cdots {\oplus }_q{a}_n)}^k\hfill \\ & \equiv \sum _{|\overrightarrow{m}|=k}\prod _{l=1}^n{\left(a_l\right)}^{m_l}{\left(\genfrac{}{}{0pt}{}{k}{\overrightarrow{m}}\right)}_q,{\oplus }_q\equiv \vee {\oplus }_q\vee {\ominus }_q\vee {⊞}_q\vee {\boxminus }_q.\hfill \end{array}$$

(17)

In formula (17) we have to multiply every term ${\left(a_l\right)}^{m_l}$ by ${(-1)}^{m_l}{q}^{\left(\genfrac{}{}{0pt}{}{m_l}{2}\right)}$ if a minus and/or a ${⊞}_q$ is preceded by $a_l$ in $F\left(k\right)$.

Definition 8.

Assume that $\overrightarrow{m}\equiv (m_1,\cdots ,m_n),m\equiv m_1+\cdots +m_n$ and $a\in {\mathbb{R}}^{★}$. The vector q-multinomial-coefficient ${\left(\genfrac{}{}{0pt}{}{a}{\overrightarrow{m}}\right)}_q^{★}$ is defined by the symmetric expression

$${\left(\genfrac{}{}{0pt}{}{a}{\overrightarrow{m}}\right)}_q^{★}\equiv \frac{{\langle -a;q\rangle }_m{(-1)}^m{q}^{-\left(\genfrac{}{}{0pt}{}{\overrightarrow{m}}{2}\right)+am}}{{\langle 1;q\rangle }_{m_1}{\langle 1;q\rangle }_{m_2}\dots {\langle 1;q\rangle }_{m_n}}.$$

(18)

Definition 9.

Let the JHC q-real numbers ${\mathbb{R}}_{{⊞}_q}$ with $n+1$ letters be defined as follows:

$${\mathbb{R}}_{{⊞}_q}\equiv \{1{\boxminus }_q{q}^a{x}_1{\boxminus }_q\cdots {\boxminus }_q{q}^a{x}_n\},{\left\{x_k\right\}}_1^n\in \mathbb{R},a\in {\mathbb{R}}^{★},\left|x_k\right|<1,0<q<1.$$

(19)

when any $x_k$ is negative, we replace ${\boxminus }_q$ with ${⊞}_q$. This means that the JHC q-real numbers in (19) are functions of $n+1$ real numbers ${\left\{x_k\right\}}_1^n,a$.

The following formula applies for a q-deformed hypercube of length 1 in ${\mathbb{R}}^n$.

Definition 10.

Assuming that the right hand side converges, and $a\in {\mathbb{R}}^{★}$:

$$\begin{array}{cc}\hfill & {(1{\boxminus }_q{q}^a{x}_1{\boxminus }_q\cdots {\boxminus }_q{q}^a{x}_n)}^{-a}\equiv {\sum _{m_1,\cdots ,m_n=0}^{\infty }\prod _{j=1}^n{(-x_j)}^{m_j}{\left(\genfrac{}{}{0pt}{}{-a}{\overrightarrow{m}}\right)}_q}^{★}{q}^{\left(\genfrac{}{}{0pt}{}{\overrightarrow{m}}{2}\right)+am}.\hfill \end{array}$$

(20)

Corollary 1.

A generalization of the q-binomial theorem. The following formula [11] applies to a q-deformed hyper-rhombus of length 1 in ${\mathbb{R}}^n$.

$$\begin{array}{cc}\hfill & {(1{\boxminus }_q{q}^a{x}_1{\boxminus }_q\cdots {\boxminus }_q{q}^a{x}_n)}^{-a}=\sum _{\overrightarrow{m}=\overrightarrow{0}}^{\overrightarrow{\infty }}\frac{{\langle a;q\rangle }_m{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle \overrightarrow{1};q\rangle }_{\overrightarrow{m}}},a\in {\mathbb{R}}^{★}.\hfill \end{array}$$

(21)

The q-real number in (19) only exists when the series (20) or (21) converge. The next formula (22) ([11], p. 183) is a q-analogue of Lauricella ([12], p. 145), ([13], p. 48 2.3.3).

Theorem 3.

A q-integral representation of the first q-Lauricella function by q-beta functions:

$$\begin{array}{cc}\hfill & B_q(\overrightarrow{b},\overrightarrow{c-b}){\mathsf{\Phi }}_A^{\left(n\right)}(a,\overrightarrow{b};\overrightarrow{c}|q;\overrightarrow{x})\hfill \\ & ={\int }_0^1\dots \left(n\right)\dots {\int }_0^1{u_1}^{b_1-1}\cdots {u_n}^{b_n-1}{(q{u}_1;q)}_{c_1-b_1-1}\hfill \\ & \cdots {(q{u}_n;q)}_{c_n-b_n-1}{(1{\boxminus }_q{q}^a{u}_1{x}_1{\boxminus }_q\cdots {\boxminus }_q{q}^a{u}_n{x}_n)}^{-a}{d}_q\left(u_1\right)\cdots d_q\left(u_n\right).\hfill \end{array}$$

(22)

Definition 11.

The convergence regions are extended depending on q, e.g., an octahedron is increased to a q-octahedron for $n=3$. The q-Lauricella functions [9] are

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_A^{\left(n\right)}(a,\overrightarrow{b};\overrightarrow{c}|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle a;q\rangle }_m{\langle \overrightarrow{b};q\rangle }_{\overrightarrow{m}}{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle \overrightarrow{c},\overrightarrow{1};q\rangle }_{\overrightarrow{m}}},|x_1|{\oplus }_q\dots {\oplus }_q|x_n|<1,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_B^{\left(n\right)}(\overrightarrow{a},\overrightarrow{b};c|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle \overrightarrow{a},\overrightarrow{b};q\rangle }_{\overrightarrow{m}}{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle c;q\rangle }_m{\langle \overrightarrow{1};q\rangle }_{\overrightarrow{m}}},max\left(\right|x_1|,\cdots ,|x_n\left|\right)<1,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_C^{\left(n\right)}(a,b;\overrightarrow{c}|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle a,b;q\rangle }_m{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle \overrightarrow{c},\overrightarrow{1};q\rangle }_{\overrightarrow{m}}},\left|\sqrt{x_1}\right|{\oplus }_q\cdots {\oplus }_q\left|\sqrt{x_n}\right|<1,\hfill \end{array}$$

(25)

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_D^{\left(n\right)}(a,\overrightarrow{b};c|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle a;q\rangle }_m{\langle \overrightarrow{b};q\rangle }_{\overrightarrow{m}}{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle c;q\rangle }_m{\langle \overrightarrow{1};q\rangle }_{\overrightarrow{m}}},max\left(\right|x_1|,\dots ,|x_n\left|\right)<1.\end{array}$$

(26)

4. Confluent Triple $\mathbf{q}$-Lauricella Functions

We refer to Exton [1], where the confluent triple hypergeometric functions were defined together with integral formulas of Euler type, as well as systems of partial differential equations. The following q-analogues have not been defined before, nor their q-difference equations, whose proofs are obvious.

Definition 12.

A q-analogue of Exton [1]. The confluent triple q-Lauricella functions are defined by

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{A,1}^{\left(3\right)}(a,a,a,b_1,b_2;\overrightarrow{c}|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle a;q\rangle }_m{\langle b_1;q\rangle }_{m_1}{\langle b_2;q\rangle }_{m_2}{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle \overrightarrow{c},\overrightarrow{1};q\rangle }_{\overrightarrow{m}}},\hfill \\ & |x_1|{\oplus }_q|x_2|<1,|(1-q)x_3|<\infty ,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{A,2}^{\left(3\right)}(a,a,a,b_1;\overrightarrow{c}|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle a;q\rangle }_m{\langle b_1;q\rangle }_{m_1}{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle \overrightarrow{c},\overrightarrow{1};q\rangle }_{\overrightarrow{m}}},\hfill \\ & |x_1|<1,|(1-q)x_2|<\infty ,|(1-q)x_3|<\infty ,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{A,3}^{\left(3\right)}(a,a,a;\overrightarrow{c}|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle a;q\rangle }_m{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle \overrightarrow{c},\overrightarrow{1};q\rangle }_{\overrightarrow{m}}},\hfill \\ & |(1-q)x_1|<\infty ,|(1-q)x_2|<\infty ,|(1-q)x_3|<\infty ,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{B,1}^{\left(3\right)}(\overrightarrow{a},b_1,b_2;c,c,c|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle \overrightarrow{a};q\rangle }_{\overrightarrow{m}}{\langle b_1;q\rangle }_{m_1}{\langle b_2;q\rangle }_{m_2}{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle c;q\rangle }_m{\langle \overrightarrow{1};q\rangle }_{\overrightarrow{m}}},\hfill \\ & |x_1|<1,|x_2|<1,|(1-q)x_3|<\infty ,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{B,2}^{\left(3\right)}(\overrightarrow{a},b_1;c,c,c|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle \overrightarrow{a};q\rangle }_{\overrightarrow{m}}{\langle b_1;q\rangle }_{m_1}{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle c;q\rangle }_m{\langle \overrightarrow{1};q\rangle }_{\overrightarrow{m}}},\hfill \\ & |x_1|<1,|(1-q)x_2|<\infty ,|(1-q)x_3|<\infty ,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{B,3}^{\left(3\right)}(a_1,a_2,b_1;c|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle a_1,b_1;q\rangle }_{m_1}{\langle a_2;q\rangle }_{m_2}{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle c;q\rangle }_m{\langle \overrightarrow{1};q\rangle }_{\overrightarrow{m}}},\hfill \\ & |x_1|<1,|(1-q)x_2|<\infty ,|{(1-q)}^2{x}_3|<\infty ,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{B,4}^{\left(3\right)}(a_1,a_2;c,c,c|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle a_1;q\rangle }_{m_1}{\langle a_2;q\rangle }_{m_2}{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle c;q\rangle }_m{\langle \overrightarrow{1};q\rangle }_{\overrightarrow{m}}},\hfill \\ & |(1-q)x_1|<\infty ,|(1-q)x_2|<\infty ,|{(1-q)}^2{x}_3|<\infty ,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{B,5}^{\left(3\right)}(a_1,a_2,b_1,b_2;c,c,c|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle a_1,b_1;q\rangle }_{m_1}{\langle a_2,b_2;q\rangle }_{m_2}{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle c;q\rangle }_m{\langle \overrightarrow{1};q\rangle }_{\overrightarrow{m}}},\hfill \\ & |x_1|<1,|x_2|<1,|{(1-q)}^2{x}_3|<\infty ,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{B,6}^{\left(3\right)}(\overrightarrow{a};c,c,c|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle \overrightarrow{a};q\rangle }_{\overrightarrow{m}}{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle c;q\rangle }_m{\langle \overrightarrow{1};q\rangle }_{\overrightarrow{m}}},\hfill \\ & |(1-q)x_1|<\infty ,|(1-q)x_2|<\infty ,|(1-q)x_3|<\infty ,\hfill \end{array}$$

(35)

$$\begin{array}{cc}& {\mathsf{\Phi }}_{D,1}^{\left(3\right)}(a,a,a,b_1,b_2;c,c,c|q;\overrightarrow{x})\equiv \sum _{\overrightarrow{m}}\frac{{\langle a;q\rangle }_m{\langle b_1;q\rangle }_{m_1}{\langle b_2;q\rangle }_{m_2}{\overrightarrow{x}}^{\overrightarrow{m}}}{{\langle c;q\rangle }_m{\langle \overrightarrow{1};q\rangle }_{\overrightarrow{m}}},\hfill \\ & |x_1|<1,|x_2|<1,|(1-q)x_3|<\infty .\hfill \end{array}$$

(36)

Theorem 4.

A q-analogue of ([1], p. 83). Put

$$\begin{array}{cc}\hfill & C\equiv {\Gamma }_q\left[\begin{array}{c}{c}_1,c_2\\ b_1,b_2,c_1-b_1,c_2-b_2\end{array}\right].\hfill \end{array}$$

(37)

A double q-integral representation of ${\mathsf{\Phi }}_{A,1}^{\left(3\right)}$.

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{A,1}^{\left(3\right)}(a,a,a,b_1,b_2;\overrightarrow{c}|q;x,y,z)=C\sum _{p=0}^{\infty }\frac{{\langle a;q\rangle }_p{z}^p}{{\langle 1,c_3;q\rangle }_p}{\int }_{\overrightarrow{0}}^{\overrightarrow{1}}{u}^{b_1-1}{v}^{b_2-1}\hfill \\ & {(qu;q)}_{c_1-b_1-1}{(qv;q)}_{c_2-b_2-1}{\left(1{\boxminus }_q{q}^{a+p}ux{\boxminus }_q{q}^{a+p}vy\right)}^{-a-p}{d}_q\left(u\right)d_q\left(v\right).\hfill \end{array}$$

(38)

Proof. 

We shall use formulas ([8], (1.46), (7.55)).

$$\begin{array}{cc}\hfill & \mathrm{RHS}=C\sum _{p=0}^{\infty }\frac{{\langle a;q\rangle }_p{z}^p}{{\langle 1,c_3;q\rangle }_p}{\int }_{\overrightarrow{0}}^{\overrightarrow{1}}{u}^{b_1-1}{v}^{b_2-1}{(qu;q)}_{c_1-b_1-1}\hfill \\ & {(qv;q)}_{c_2-b_2-1}\sum _{m,n=0}^{\infty }\frac{{\langle a+p;q\rangle }_{m+n}}{{\langle 1;q\rangle }_m{\langle 1;q\rangle }_n}{\left(ux\right)}^m{\left(vy\right)}^n{d}_q\left(u\right)d_q\left(v\right)\hfill \\ & =C\sum _{m,n,p=0}^{\infty }\frac{{\langle a;q\rangle }_{m+n+p}{x}^m{y}^n{z}^p}{{\langle 1;q\rangle }_m{\langle 1;q\rangle }_n{\langle 1,c_3;q\rangle }_p}\hfill \\ & {\Gamma }_q\left[\begin{array}{c}{c}_1-b_1,c_2-b_2,b_1+m,b_2+n\\ c_1+m,c_2+n\end{array}\right]=\mathrm{LHS}.\hfill \end{array}$$

(39)

□

Theorem 5.

A q-analogue of ([1], p. 85). A Euler q-integral representation of ${\mathsf{\Phi }}_{A,2}^{\left(3\right)}$.

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{A,2}^{\left(3\right)}(a,a,a,b_1;\overrightarrow{c}|q;x,y,z)={\Gamma }_q\left[\begin{array}{c}{c}_1\\ b_1,c_1-b_1\end{array}\right]{\int }_0^1{t}^{b_1-1}\hfill \\ & \frac{{(qt;q)}_{c_1-b_1-1}}{{(xt;q)}_a}\sum _{n,p=0}^{\infty }\frac{{\langle a;q\rangle }_{n+p}{y}^n{z}^p}{{\langle 1,c_2;q\rangle }_n{\langle 1,c_3;q\rangle }_p{(xt{q}^a;q)}_{n+p}}{d}_q\left(t\right).\hfill \end{array}$$

(40)

Proof. 

Use formula ([8], (7.50)). □

Theorem 6.

A Euler q-integral representation of ${\mathsf{\Phi }}_{D,1}^{\left(3\right)}$.

$$\begin{array}{cc}\hfill & {\mathsf{\Phi }}_{D,1}^{\left(3\right)}(a,a,a,b_1,b_2;c,c,c|q;x,y,z)\hfill \\ & ={\Gamma }_q\left[\begin{array}{c}c\\ a,c-a\end{array}\right]\sum _{p=0}^{\infty }\frac{z^p}{{\langle 1;q\rangle }_p}{\int }_0^1{t}^{a+p-1}\frac{{(qt;q)}_{c-a-1}}{{(xt;q)}_{b_1}{(yt;q)}_{b_2}}{d}_q\left(t\right).\hfill \end{array}$$

(41)

Proof. 

Use formula ([8], (10.104)). □

We now turn to the corresponding canonical q-difference equations. Of course, there are many forms of these equations.

Theorem 7.

The canonical q-difference equations for the confluent function ${\mathsf{\Phi }}_{\mathrm{A},1}^{\left(3\right)}$ above, a q-analogue of ([1], p. 87), are

$$\left\{\begin{array}{c}-{\mathbf{x}}_{\mathbf{1}}{\{{\theta }_{q,1}+b_1\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+a\}}_q+{\left\{{\theta }_{q,1}\right\}}_q{\{{\theta }_{q,1}+c_1-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{2}}{\{{\theta }_{q,2}+b_2\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+a\}}_q+{\left\{{\theta }_{q,2}\right\}}_q{\{{\theta }_{q,2}+c_2-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{3}}{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+a\}}_q+(1-q){\left\{{\theta }_{q,3}\right\}}_q{\{{\theta }_{q,3}+c_3-1\}}_q=0.\hfill \end{array}\right.$$

(42)

Theorem 8.

The canonical q-difference equations for the confluent function ${\mathsf{\Phi }}_{\mathrm{A},2}^{\left(3\right)}$ above, a q-analogue of ([1], p. 87), are

$$\left\{\begin{array}{c}-{\mathbf{x}}_{\mathbf{1}}{\{{\theta }_{q,1}+b_1\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+a\}}_q+{\left\{{\theta }_{q,1}\right\}}_q{\{{\theta }_{q,1}+c_1-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{2}}{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+a\}}_q+(1-q){\left\{{\theta }_{q,2}\right\}}_q{\{{\theta }_{q,2}+c_2-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{3}}{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+a\}}_q+(1-q){\left\{{\theta }_{q,3}\right\}}_q{\{{\theta }_{q,3}+c_3-1\}}_q=0.\hfill \end{array}\right.$$

(43)

Theorem 9.

The canonical q-difference equations for the confluent function ${\mathsf{\Phi }}_{\mathrm{A},3}^{\left(3\right)}$ above, a q-analogue of ([1], p. 87), are

$$\left\{\begin{array}{c}-{\mathbf{x}}_{\mathbf{1}}{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+a\}}_q+(1-q){\left\{{\theta }_{q,1}\right\}}_q{\{{\theta }_{q,1}+c_1-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{2}}{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+a\}}_q+(1-q){\left\{{\theta }_{q,2}\right\}}_q{\{{\theta }_{q,2}+c_2-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{3}}{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+a\}}_q+(1-q){\left\{{\theta }_{q,3}\right\}}_q{\{{\theta }_{q,3}+c_3-1\}}_q=0.\hfill \end{array}\right.$$

(44)

Theorem 10.

The canonical q-difference equations for the confluent function ${\mathsf{\Phi }}_{\mathrm{B},1}^{\left(3\right)}$ above, a q-analogue of ([1], p. 87), are

$$\left\{\begin{array}{c}-{\mathbf{x}}_{\mathbf{1}}{\{{\theta }_{q,1}+a_1\}}_q{\{{\theta }_{q,1}+b_1\}}_q+{\left\{{\theta }_{q,1}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{2}}{\{{\theta }_{q,2}+a_2\}}_q{\{{\theta }_{q,2}+b_2\}}_q+{\left\{{\theta }_{q,2}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{3}}{\{{\theta }_{q,3}+a_3\}}_q+(1-q){\left\{{\theta }_{q,3}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0.\hfill \end{array}\right.$$

(45)

Theorem 11.

The canonical q-difference equations for the confluent function ${\mathsf{\Phi }}_{\mathrm{B},2}^{\left(3\right)}$ above, a q-analogue of ([1], p. 87), are

$$\left\{\begin{array}{c}-{\mathbf{x}}_{\mathbf{1}}{\{{\theta }_{q,1}+a_1\}}_q{\{{\theta }_{q,1}+b_1\}}_q+{\left\{{\theta }_{q,1}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{2}}{\{{\theta }_{q,2}+a_2\}}_q+(1-q){\left\{{\theta }_{q,2}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{3}}{\{{\theta }_{q,3}+a_3\}}_q+(1-q){\left\{{\theta }_{q,3}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0.\hfill \end{array}\right.$$

(46)

Theorem 12.

The canonical q-difference equations for the confluent function ${\mathsf{\Phi }}_{\mathrm{B},3}^{\left(3\right)}$ above, a q-analogue of ([1], p. 88), are

$$\left\{\begin{array}{c}-{\mathbf{x}}_{\mathbf{1}}{\{{\theta }_{q,1}+a_1\}}_q{\{{\theta }_{q,1}+b_1\}}_q+{\left\{{\theta }_{q,1}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{2}}{\{{\theta }_{q,2}+a_2\}}_q+(1-q){\left\{{\theta }_{q,2}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{3}}+{(1-q)}^2{\left\{{\theta }_{q,3}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0.\hfill \end{array}\right.$$

(47)

Theorem 13.

The canonical q-difference equations for the confluent function ${\mathsf{\Phi }}_{\mathrm{B},4}^{\left(3\right)}$ above, a q-analogue of ([1], p. 88), are

$$\left\{\begin{array}{c}-{\mathbf{x}}_{\mathbf{1}}{\{{\theta }_{q,1}+a_1\}}_q+(1-q){\left\{{\theta }_{q,1}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{2}}{\{{\theta }_{q,2}+a_2\}}_q+(1-q){\left\{{\theta }_{q,2}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{3}}+{(1-q)}^2{\left\{{\theta }_{q,3}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0.\hfill \end{array}\right.$$

(48)

Theorem 14.

The canonical q-difference equations for the confluent function ${\mathsf{\Phi }}_{\mathrm{B},5}^{\left(3\right)}$ above, a q-analogue of ([1], p. 88), are

$$\left\{\begin{array}{c}-{\mathbf{x}}_{\mathbf{1}}{\{{\theta }_{q,1}+a_1\}}_q{\{{\theta }_{q,1}+b_1\}}_q+{\left\{{\theta }_{q,1}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{2}}{\{{\theta }_{q,2}+a_2\}}_q{\{{\theta }_{q,2}+b_2\}}_q+{\left\{{\theta }_{q,2}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{3}}+{(1-q)}^2{\left\{{\theta }_{q,3}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0.\hfill \end{array}\right.$$

(49)

Theorem 15.

The canonical q-difference equations for the confluent function ${\mathsf{\Phi }}_{\mathrm{B},6}^{\left(3\right)}$ above, a q-analogue of ([1], p. 88), are

$$\left\{\begin{array}{c}-{\mathbf{x}}_{\mathbf{1}}{\{{\theta }_{q,1}+a_1\}}_q+(1-q){\left\{{\theta }_{q,1}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{2}}{\{{\theta }_{q,2}+a_2\}}_q+(1-q){\left\{{\theta }_{q,2}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{3}}{\{{\theta }_{q,3}+a_3\}}_q+(1-q){\left\{{\theta }_{q,3}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0.\hfill \end{array}\right.$$

(50)

Theorem 16.

The canonical q-difference equations for the confluent function ${\mathsf{\Phi }}_{\mathrm{D},1}^{\left(3\right)}$ above, a q-analogue of ([1], p. 88), are

$$\left\{\begin{array}{c}-{\mathbf{x}}_{\mathbf{1}}{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+a\}}_q{\{{\theta }_{q,1}+b_1\}}_q+{\left\{{\theta }_{q,1}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{2}}{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+a\}}_q{\{{\theta }_{q,2}+b_2\}}_q+{\left\{{\theta }_{q,2}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0,\hfill \\ -{\mathbf{x}}_{\mathbf{3}}{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+a\}}_q+(1-q){\left\{{\theta }_{q,3}\right\}}_q{\{{\theta }_{q,1}+{\theta }_{q,2}+{\theta }_{q,3}+c-1\}}_q=0.\hfill \end{array}\right.$$

(51)

We note that for each confluence we multiply with powers of $(1-q)$.

5. A Summation Formula for the Second $\mathit{q}$-Appell Function

The aim of this section is to prove a summation formula for the second q-Appell function, which is used to prove transformation formulas for the q-Horn functions in the next section. We start with a lemma for q-shifted factorials.

Lemma 1.

$$\frac{{\langle e-a-k;q\rangle }_n{\langle 1+a-n-e;q\rangle }_k}{{\langle e-a;q\rangle }_n{\langle 1+a-e;q\rangle }_k}=q^{-nk}.$$

(52)

Proof. 

We use formulas ([8], 2x(6.15)) and ([8], (6.13)).

$$\begin{array}{cc}& \mathrm{LHS}=\frac{1}{{\langle e-a;q\rangle }_{n-k}{\langle 1+a-e;q\rangle }_{k-n}}{(-1)}^{n-k}\hfill \\ & \mathrm{QE}\left(\left(\genfrac{}{}{0pt}{}{n}{2}\right)+\left(\genfrac{}{}{0pt}{}{k}{2}\right)+n(e-a-k)+k(1+a-e-n)\right)=\mathrm{RHS}.\hfill \end{array}$$

(53)

□

Theorem 17.

A summation formula for the second q-Appell function, a q-analogue of Lal ([14], (2.1)).

$${\mathsf{\Phi }}_2(a;-m,-n;1+a-e,e|q;q,q)=\frac{{\langle e-a;q\rangle }_{n-m}}{{\langle e;q\rangle }_{n-m}}{q}^{2na}.$$

(54)

Proof. 

We shall use formula ([8], 2x(6.15)) and the second q-Vandermonde theorem ([8], (7.68)) twice together with (52).

$$\begin{array}{cc}\hfill & \mathrm{LHS}=\sum _{k=0}^m\frac{{\langle a,-m;q\rangle }_k}{{\langle 1,1+a-e;q\rangle }_k}{q}^k{}_2{\mathsf{\Phi }}_1(a+k,-n;e|q;q)\hfill \\ & =\sum _{k=0}^m\frac{{\langle a,-m;q\rangle }_k}{{\langle 1,1+a-e;q\rangle }_k}{q}^k\frac{\langle {e-a-k;q\rangle }_n}{{\langle e;q\rangle }_n}{q}^{n(a+k)}\hfill \\ & =\frac{\langle {e-a;q\rangle }_n}{\langle {e;q\rangle }_n}{q}^{na}{}_2{\mathsf{\Phi }}_1(a,-m;1+a-e-n|q;q)\hfill \\ & =\frac{{\langle e-a;q\rangle }_n{\langle 1-n-e;q\rangle }_m}{{\langle e;q\rangle }_n{\langle 1+a-n-e;q\rangle }_m}{q}^{a(m+n)}=\mathrm{RHS}.\hfill \end{array}$$

(55)

□

6. Transformation Formulas for the $\mathbf{q}$-Horn Functions

For all the transformation formulas in this section, the reader can easily convince himself that the limit $q\to 1$ in our formulas is the corresponding original Horn function transformation formula. As usual, the q-deformed formulas are nested sums; there are many forms of these q-analogues.

Theorem 18.

A q-analogue of the Lal ([14], (3.1)) transformation formula for the Horn function G${}_1$.

$$\begin{array}{cc}\hfill & \sum _{m,n=0}^{\infty }\frac{{\langle a;q\rangle }_{m+n}{\langle c-b;q\rangle }_{n-m}{\langle 1-c;q\rangle }_{m-n}}{{\langle 1;q\rangle }_m{\langle 1;q\rangle }_n}{x}^m{y}^n\hfill \\ & \mathrm{QE}\left(-\left(\genfrac{}{}{0pt}{}{m-n}{2}\right)+2nb+(m-n)(c-1)\right)\hfill \\ & =\sum _{p,r=0}^{\infty }\frac{{\langle a,b;q\rangle }_{p+r}{x}^p{y}^r}{{\langle 1,1+b-c;q\rangle }_p{\langle 1,c;q\rangle }_r}\mathrm{QE}\left(-\left(\genfrac{}{}{0pt}{}{p}{2}\right)-\left(\genfrac{}{}{0pt}{}{r}{2}\right)-pm-rn\right)\hfill \\ & q^{m+n}{\left(1{\boxminus }_q{q}^{a+p+r}(-x){\boxminus }_q{q}^{a+p+r}(-y)\right)}^{-a-p-r}.\hfill \end{array}$$

(56)

Proof. 

We shall use the formulas ([8], (6.13), (1.6), (6.20)), together with (54) and (21).

$$\begin{array}{cc}\hfill & \mathrm{LHS}=\sum _{m,n=0}^{\infty }\frac{{\langle a;q\rangle }_{m+n}{\langle c-b;q\rangle }_{n-m}}{{\langle 1;q\rangle }_m{\langle 1;q\rangle }_n{\langle c;q\rangle }_{n-m}}{(-x)}^m{(-y)}^n{q}^{2nb}\hfill \\ \hfill & =\sum _{m,n=0}^{\infty }\frac{{\langle a;q\rangle }_{m+n}{(-x)}^m{(-y)}^n}{{\langle 1;q\rangle }_m{\langle 1;q\rangle }_n}{\mathsf{\Phi }}_2(b;-m,-n;1+b-c,c|q;q,q)\hfill \\ \hfill & =\sum _{m,n=0}^{\infty }\sum _{p,r=0}^{m,n}\frac{{\langle a;q\rangle }_{m+n}{\langle b;q\rangle }_{p+r}{\langle -m;q\rangle }_p{\langle -n;q\rangle }_r}{{\langle 1;q\rangle }_m{\langle 1;q\rangle }_n{\langle 1,1+b-c;q\rangle }_p{\langle 1,c;q\rangle }_r}{(-x)}^m{(-y)}^n{q}^{m+n}\hfill \\ \hfill & =\sum _{m,n,p,r=0}^{\infty }\frac{{\langle a;q\rangle }_{m+n+p+r}{\langle b;q\rangle }_{p+r}{\langle -m-p;q\rangle }_p{\langle -n-r;q\rangle }_r}{{\langle 1;q\rangle }_{m+p}{\langle 1;q\rangle }_{n+r}{\langle 1,1+b-c;q\rangle }_p{\langle 1,c;q\rangle }_r}\hfill \\ & {(-x)}^{m+p}{(-y)}^{n+r}{q}^{m+n+p+r}=\mathrm{RHS}.\hfill \end{array}$$

(57)

□

Theorem 19.

A q-analogue of the Lal ([14], (3.2)) transformation formula for the Horn function G${}_2$.

$$\begin{array}{cc}\hfill & \sum _{m,n=0}^{\infty }\frac{{\langle a;q\rangle }_m{\langle a^{\prime };q\rangle }_n{\langle c-b;q\rangle }_{n-m}{\langle 1-c;q\rangle }_{m-n}}{{\langle 1;q\rangle }_m{\langle 1;q\rangle }_n}{x}^m{y}^n\hfill \\ & \mathrm{QE}\left(-\left(\genfrac{}{}{0pt}{}{m-n}{2}\right)+2nb+(m-n)(c-1)\right)\hfill \\ & =\sum _{p,r=0}^{\infty }\frac{{\langle a;q\rangle }_p{\langle a^{\prime };q\rangle }_r{\langle b;q\rangle }_{p+r}{x}^p{y}^r}{{\langle 1,1+b-c;q\rangle }_p{\langle 1,c;q\rangle }_r}\mathrm{QE}\left(-\left(\genfrac{}{}{0pt}{}{p}{2}\right)-\left(\genfrac{}{}{0pt}{}{r}{2}\right)-pm-rn\right)\hfill \\ & q^{m+n}{\left(1{\boxminus }_q{q}^{a+p}(-x)\right)}^{-a-p}{\left(1{\boxminus }_q{q}^{a+r}(-y)\right)}^{-a-r}.\hfill \end{array}$$

(58)

Proof. 

We shall use the formulas ([8], (6.13), (1.6), (6.20)), together with (54).

$$\begin{array}{cc}& \mathrm{LHS}=\sum _{m,n=0}^{\infty }\frac{{\langle a;q\rangle }_m{\langle a^{\prime };q\rangle }_n{\langle c-b;q\rangle }_{n-m}}{{\langle 1;q\rangle }_m{\langle 1;q\rangle }_n{\langle c;q\rangle }_{n-m}}{(-x)}^m{(-y)}^n{q}^{2nb}\hfill \\ & =\sum _{m,n=0}^{\infty }\frac{{\langle a;q\rangle }_m{\langle a^{\prime };q\rangle }_n{(-x)}^m{(-y)}^n}{{\langle 1;q\rangle }_m{\langle 1;q\rangle }_n}{\mathsf{\Phi }}_2(b;-m,-n;1+b-c,c|q;q,q)\hfill \\ & =\sum _{m,n=0}^{\infty }\sum _{p,r=0}^{m,n}\frac{{\langle a;q\rangle }_m{\langle a^{\prime };q\rangle }_n{\langle b;q\rangle }_{p+r}{\langle -m;q\rangle }_p{\langle -n;q\rangle }_r}{{\langle 1;q\rangle }_m{\langle 1;q\rangle }_n{\langle 1,1+b-c;q\rangle }_p{\langle 1,c;q\rangle }_r}{(-x)}^m{(-y)}^n\hfill \\ & =\sum _{m,n,p,r=0}^{\infty }\frac{{\langle a;q\rangle }_{m+p}{\langle a^{\prime };q\rangle }_{n+r}{\langle b;q\rangle }_{p+r}{\langle -m-p;q\rangle }_p{\langle -n-r;q\rangle }_r}{{\langle 1;q\rangle }_{m+p}{\langle 1;q\rangle }_{n+r}{\langle 1,1+b-c;q\rangle }_p{\langle 1,c;q\rangle }_r}\hfill \\ & {(-x)}^{m+p}{(-y)}^{n+r}=\mathrm{RHS}.\hfill \end{array}$$

(59)

□

Theorem 20.

A q-analogue of the Qureshi–Pathan ([15], (1.6)) transformation formula for the Horn function H${}_1$.

$$\begin{array}{cc}\hfill & \sum _{m,n=0}^{\infty }\frac{{\langle e-a;q\rangle }_{m-n}{\langle 1-a;q\rangle }_{m+n}{\langle a;q\rangle }_n}{{\langle 1,e;q\rangle }_m{\langle 1;q\rangle }_n{(-x;q)}_m}{y}^m{x}^n{(-x;q)}_{1-a+m}\hfill \\ & \times \mathrm{QE}\left(-\left(\genfrac{}{}{0pt}{}{n}{2}\right)-n(1-e-m+a)\right)\hfill \\ & =\sum _{m,n=0}^{\infty }\frac{{\langle 1-a;q\rangle }_{m+n}{\langle e-a;q\rangle }_{m-n}{\langle 1-e;q\rangle }_{n-m}}{{\langle 1;q\rangle }_m{\langle 1;q\rangle }_n{(-x;q)}_m{(-x{q}^{1-a+m};q)}_n}\hfill \\ & {(-y)}^m{(-x)}^n\mathrm{QE}\left(-\left(\genfrac{}{}{0pt}{}{m}{2}\right)+n(-1+e+m+a)-me\right).\hfill \end{array}$$

(60)

Proof. 

We shall use the formulas ([8], (6.13), (6.15), (7.62)).

$$\begin{array}{cc}& \mathrm{LHS}=\sum _{m,n=0}^{\infty }\frac{{\langle 1-a;q\rangle }_{m+n}{\langle e-a;q\rangle }_m{\langle a;q\rangle }_n}{{\langle 1,e;q\rangle }_m{\langle 1,1-e-m+a;q\rangle }_n{(-x;q)}_m}{y}^m{(-x)}^n{(-x;q)}_{1-a+m}\hfill \\ & =\sum _{m=0}^{\infty }\frac{{\langle 1-a,e-a;q\rangle }_m{y}^m}{{\langle 1,e;q\rangle }_m{(-x;q)}_m}\hfill \\ & {}_2{\mathsf{\Phi }}_2(1-a+m,1-e-m;1-e-m+a|q;-x{q}^a\left|\right|-;-x{q}^{1-a+m})\hfill \\ & =\mathrm{RHS}.\hfill \end{array}$$

(61)

□

7. Discussion

One could compare the q-integrals and q-difference equations with the recent paper [7]. We were unable to verify the Formulas ([15], (1.7)–(1.9)) according to Per Karlssons review. The reason that q-integral expressions for the confluent ${\mathsf{\Phi }}_{B,i}^{\left(3\right)}$ functions could not be found, was because of the complicated integral formulas for ${\mathsf{\Phi }}_B^{\left(n\right)}$ functions, which could not be q-deformed.

8. Conclusions

We have shown that our umbral calculus can be extended to any number of variables, and our formulas for lesser number of variables can be used in the proofs. The reason that q-real numbers can be used for our purposes is the symmetric character of the first and third q-real numbers. Unlike the q-Laplace integrals, balanced ${\Gamma }_q$-functions occur as prefactors in the Euler integral formulas.

For the confluent triple q-Lauricella functions, the convergence regions always increase for the variable, whose index has confluence.