*Article* **On the Triple Lauricella–Horn–Karlsson** *q***-Hypergeometric Functions**

#### **Thomas Ernst**

Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 Uppsala, Sweden; thomas@math.uu.se Received: 29 May 2020; Accepted: 2 July 2020; Published: 31 July 2020

**Abstract:** The Horn–Karlsson approach to find convergence regions is applied to find convergence regions for triple *q*-hypergeometric functions. It turns out that the convergence regions are significantly increased in the *q*-case; just as for *q*-Appell and *q*-Lauricella functions, additions are replaced by Ward *q*-additions. Mostly referring to Krishna Srivastava 1956, we give *q*-integral representations for these functions.

**Keywords:** triple *q*-hypergeometric function; convergence region; Ward *q*-addition; *q*-integral representation

#### **MSC:** 33D70; 33C65

#### **1. Introduction**

This is part of a series of papers about *q*-integral representations of *q*-hypergeometric functions. The first paper [1] was about *q*-hypergeometric transformations involving *q*-integrals. Then followed [2], where Euler *q*-integral representations of *q*-Lauricella functions in the spirit of Koschmieder were presented. Furthermore, in [3], Eulerian *q*-integrals for single and multiple *q*-hypergeometric series were found. However, this subject is by no means exhausted, and in the same proceedings, [4], concise proofs for *q*-analogues of Eulerian integral formulas for general *q*-hypergeometric functions corresponding to Erdélyi, and for two of Srivastavas triple hypergeometric functions were given. Finally, in [5], single and multiple *q*-Eulerian integrals in the spirit of Exton, Driver, Johnston, Pandey, Saran and Erdélyi are presented. All proofs use the *q*-beta integral method.

The history of the subject in this article started in 1889 when Horn [6] investigated the domain of convergence for double and triple *q*-hypergeometric functions. He invented an ingenious geometric construction with five sets of convergence regions in three dimensions which was successfully used by Karlsson [7] in 1974 to explicitly state the convergence regions for the known functions of three variables. We adapt this approach to the *q*-case, by replacing additions by *q*-additions and exactly stating the convergence sets for each function. Obviously combinations of the *q*-deformed rhombus in dimension three appear several times. It is not possible to depict the *q*-additions in diagrams, not even in dimension two; they depend on the parameter *q*. We recall Karlssons paper, which seems to have fallen into oblivion. We give proofs for all the convergence regions, and our proofs also work for Karlssons equations by putting *q* = 1.

Saran [8], followed by Exton [9] gave less correct convergence criteria. By giving *q*-integral representations for these functions, we also correct and give proofs for the formulas in K.J. Srivastava [10] (not Hari Srivastava). He did not give many proofs, and our proofs also work for his equations by putting *q* = 1.

#### **2. Definitions**

**Definition 1.** *We define 10 q-analogues of the three-variable Lauricella–Saran functions of three variables plus two G-functions. Each function is defined by*

$$F \equiv \sum\_{m,n,p=0}^{+\infty} \Psi \frac{x^m y^n z^p}{\langle 1;q\rangle\_m \langle 1;q\rangle\_n \langle 1;q\rangle\_p}.\tag{1}$$

*As a result of lack of space, for every row, we first give the generic name, the function parameters, followed by the corresponding* Ψ *according to (1).*


In the whole paper, *Aq*,*m*,*n*,*<sup>p</sup>* denotes the coefficient of *xmynzp* for the respective function. In the following, we follow the notation in Karlsson [7].

Discarding possible discontinuities, we introduce the following three rational functions:

$$\begin{aligned} \Psi\_1(m, n, p) &\equiv \lim\_{\varepsilon \to +\infty} \frac{A\_{1, \varepsilon m + 1, \varepsilon n, \varepsilon p}}{A\_{\varepsilon m, \varepsilon n, \varepsilon p}}, \ m > 0, \ n \ge 0, \ p \ge 0, \\\Psi\_2(m, n, p) &\equiv \lim\_{\varepsilon \to +\infty} \frac{A\_{1, \varepsilon m, \varepsilon n + 1, \varepsilon p}}{A\_{\varepsilon m, \varepsilon n, \varepsilon p}}, \ m \ge 0, \ n > 0, \ p \ge 0, \\\Psi\_3(m, n, p) &\equiv \lim\_{\varepsilon \to +\infty} \frac{A\_{1, \varepsilon m, \varepsilon n, \varepsilon p + 1}}{A\_{\varepsilon m, \varepsilon n, \varepsilon p}}, \ m \ge 0, \ n \ge 0, \ p > 0. \end{aligned} \tag{2}$$

For 0 < *q* < 1 fixed, exactly as in Karlsson [7], construct the following subsets of R<sup>3</sup> +:

$$\begin{aligned} \mathbb{C}\_{q} & \equiv \{ (r, s, t) \, | \, 0 < r < |\mathbb{Y}\_{1}(1, 0, 0)|^{-1} \land 0 < s < |\mathbb{Y}\_{2}(0, 1, 0)|^{-1} \land \\ & \land 0 < t < |\mathbb{Y}\_{3}(0, 0, 1)|^{-1} \}, \end{aligned} \tag{3}$$

$$X\_q \equiv \{(r, s, t) \mid \forall (n, p) \in \mathbb{R}\_+^2 : 0 < s < |\Psi\_2(0, n, p)|^{-1} \lor 0 < t < |\Psi\_3(0, n, p)|^{-1}\},\tag{4}$$

$$\mathcal{Y}\_{\mathfrak{q}} \equiv \{(r, s, t) \mid \forall (m, p) \in \mathbb{R}\_+^2 : 0 < r < |\Psi\_1(m, 0, p)|^{-1} \lor 0 < t < |\Psi\_3(m, 0, p)|^{-1}\},\tag{5}$$

$$Z\_{\eta} \equiv \{(r, s, t) \mid \forall (m, n) \in \mathbb{R}\_+^2 : 0 < r < \left| \Psi\_1(m, n, 0) \right|^{-1} \lor 0 < s < \left| \Psi\_2(m, n, 0) \right|^{-1} \},\tag{6}$$

$$\begin{aligned} E\_q &\equiv \{(r, s, t) | \forall (m, n, p) \in \mathbb{R}\_+^3 : 0 < r < |\Psi\_1(m, n, p)|^{-1} \lor \\ &\lor 0 < s < |\Psi\_2(m, n, p)|^{-1} \lor 0 < t < |\Psi\_3(m, n, p)|^{-1} \}, \end{aligned} \tag{7}$$

$$D'\_q \equiv E\_q \cap X\_q \cap Y\_q \cap Z\_q \cap \mathbb{C}\_{q\text{'}} \tag{8}$$

Then let *Dq* <sup>⊆</sup> (R<sup>+</sup> ∪ {0})<sup>3</sup> denote the union of *<sup>D</sup> <sup>q</sup>* and its projections onto the coordinate planes. Horn's theorem adapted to the *q*-case then states that the region *Dq* is the representation in the absolute octant of the convergence region in *C*<sup>3</sup> *<sup>q</sup>* . We will describe *D <sup>q</sup>*, and *Dq* by that part *Sq* of *∂D <sup>q</sup>* which is not contained in coordinate planes.

**Theorem 1.** *For every row, we first give the generic name, D q, followed by the corresponding q-Cartesian equations of Sq.*


The idea is to follow Karlsson's proofs and then replace the additions by the respective *q*-additions. This gives identical convergence regions as for *q*-Appell and *q*-Lauricella functions. For each function, for didactic reasons, we first compute the quotient of corresponding coefficients.

**Proof.** For the notation we refer to [2]. Consider the function ΦE. We have

$$\begin{aligned} \frac{A\_{q,m+1,n,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_1+m+n+p, \beta\_1+m; q \rangle\_1}{\langle \gamma\_1+m, 1+m; q \rangle\_1},\\ \frac{A\_{q,m,n+1,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_1+m+n+p, \beta\_2+n+p; q \rangle\_1}{\langle \gamma\_2+n, 1+n; q \rangle\_1},\\ \frac{A\_{q,m,n,p+1}}{A\_{q,m,n,p}} &= \frac{\langle a\_1+m+n+p, \beta\_2+n+p; q \rangle\_1}{\langle \gamma\_3+p, 1+p; q \rangle\_1}.\end{aligned} \tag{9}$$

Then we have

$$\begin{aligned} \mathbf{C}\_{q} &= \{(r,s,t) \mid 0 < r < 1 \land 0 < s < 1 \land 0 < t < 1\} \\ X\_{q} &= \{(r,s,t) \mid 0 < s < \left(\frac{n}{n+p}\right)^{2} \land 0 < t < \left(\frac{p}{n+p}\right)^{2}\} \\ Y\_{q} &= \{(r,s,t) \mid 0 < r < \frac{m}{m+p} \land 0 < t < \frac{p}{m+p}\} \\ Z\_{q} &= \{(r,s,t) \mid 0 < r < \frac{m}{m+n} \land 0 < s < \frac{n}{m+n}\} \\ E\_{q} &= \{(r,s,t) \mid 0 < r < \frac{m}{m+n+p} \land 0 < s < \frac{n^{2}}{(m+n+p)(n+p)} \land \\ &\land 0 < t < \frac{p^{2}}{(m+n+p)(n+p)}\}. \end{aligned} \tag{10}$$

We have convergence domain *r* ⊕*<sup>q</sup> s* ⊕*<sup>q</sup> t* ⊕*<sup>q</sup>* 2 √*s* √*t n* < 1.

In the following, we do not write regions which are obviously bounded by 0 < *x* < 1. Consider the function ΦF. We have

$$\begin{split} \frac{A\_{q,m+1,n,p}}{A\_{q,m,n,p}} &= \frac{\langle \alpha\_1 + m + n + p, \beta\_1 + m + p; q \rangle\_1}{\langle \gamma\_1 + m, 1 + m; q \rangle\_1}, \\ \frac{A\_{q,m,n+1,p}}{A\_{q,m,n,p}} &= \frac{\langle \alpha\_1 + m + n + p, \beta\_2 + n; q \rangle\_1}{\langle \gamma\_2 + n + p, 1 + n; q \rangle\_1}, \\ \frac{A\_{q,m,n,p+1}}{A\_{q,m,n,p}} &= \frac{\langle \alpha\_1 + m + n + p, \beta\_1 + m + p; q \rangle\_1}{\langle \gamma\_2 + n + p, 1 + p; q \rangle\_1}. \end{split} \tag{11}$$

Then we have the following regions

$$\begin{aligned} Y\_{\overline{q}} &= \{(r, s, t) \mid 0 < r < \left(\frac{m}{m+p}\right)^2 \land 0 < t < \left(\frac{p}{m+p}\right)^2\} \\ Z\_{\overline{q}} &= \{(r, s, t) \mid 0 < r < \frac{m}{m+n} \land 0 < s < \frac{n}{m+n}\} \\ E\_q &= \{(r, s, t) \mid 0 < r < \frac{m^2}{(m+n+p)(m+p)} \land 0 < s < \frac{n+p}{m+n+p} \land \\ &\land 0 < t < \frac{(n+p)p}{(m+n+p)(m+p)}\}.\end{aligned} \tag{12}$$

We have convergence domain *rs <sup>t</sup>* < 1. Consider the function ΦG. We have

$$\begin{split} \frac{A\_{q,m+1,n,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_1+m+n+p,\beta\_1+m;q\rangle\_1}{\langle \gamma\_1+m,1+m;q\rangle\_1},\\ \frac{A\_{q,m,n+1,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_1+m+n+p,\beta\_2+n;q\rangle\_1}{\langle \gamma\_2+n+p,1+n;q\rangle\_1},\\ \frac{A\_{q,m,n,p+1}}{A\_{q,m,n,p}} &= \frac{\langle a\_1+m+n+p,\beta\_3+p;q\rangle\_1}{\langle \gamma\_2+n+p,1+p;q\rangle\_1}.\end{split} \tag{13}$$

Then we have the following regions

$$\begin{aligned} \mathbb{Y}\_{\emptyset} &= \{(r, s, t) \mid 0 < r < \frac{m}{m+p} \land 0 < t < \frac{p}{m+p}\} \\ \mathbb{Z}\_{\emptyset} &= \{(r, s, t) \mid 0 < r < \frac{m}{m+n} \land 0 < s < \frac{n}{m+n}\} \\ \mathbb{Z}\_{q} &= \{(r, s, t) \mid 0 < r < \frac{m}{m+n+p} \land 0 < s < \frac{n+p}{m+n+p} \land \\ \land 0 < t < \frac{n+p}{m+n+p}\} .\end{aligned} \tag{14}$$

We have convergence domain *r* ⊕*<sup>q</sup> t* < 1, *r* ⊕*<sup>q</sup> s* < 1. Consider the function ΦK. We have

$$\begin{split} \frac{A\_{q,m+1,n,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_1+m,\beta\_1+m+p;q\rangle\_1}{\langle \gamma\_1+m,1+m;q\rangle\_1},\\ \frac{A\_{q,m,n+1,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_2+n+p,\beta\_2+n;q\rangle\_1}{\langle \gamma\_2+n,1+n;q\rangle\_1},\\ \frac{A\_{q,m,n,p+1}}{A\_{q,m,n,p}} &= \frac{\langle a\_2+n+p,\beta\_1+m+p;q\rangle\_1}{\langle \gamma\_3+p,1+p;q\rangle\_1}. \end{split} \tag{15}$$

Then we have the following regions

$$\begin{aligned} X\_{\emptyset} &= \{(r, s, t) \mid 0 < s < \frac{n}{n+p} \land 0 < t < \frac{p}{n+p}\} \\ Y\_q &= \{(r, s, t) \mid 0 < r < \frac{m}{m+p} \land 0 < t < \frac{p}{m+p}\} \\ E\_q &= \{(r, s, t) \mid 0 < r < \frac{m}{m+p} \land 0 < s < \frac{n}{n+p} \land \\ &\land 0 < t < \frac{p^2}{(m+p)(n+p)}\}. \end{aligned} \tag{16}$$

We have convergence domain *rs <sup>t</sup>* < 1. Consider the function ΦM. We have

$$\begin{split} \frac{A\_{q,m+1,n,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_1+m,\beta\_1+m+p;q\rangle\_1}{\langle \gamma\_1+m,1+m;q\rangle\_1},\\ \frac{A\_{q,m,n+1,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_2+n+p,\beta\_2+n;q\rangle\_1}{\langle \gamma\_2+n+p,1+n;q\rangle\_1},\\ \frac{A\_{q,m,n,p+1}}{A\_{q,m,n,p}} &= \frac{\langle a\_2+n+p,\beta\_1+m+p;q\rangle\_1}{\langle \gamma\_2+n+p,1+p;q\rangle\_1}. \end{split} \tag{17}$$

We have the following regions

$$\begin{aligned} Y\_q &= \{(r, s, t) \mid 0 < r < \frac{m}{m+p} \land 0 < t < \frac{p}{m+p}\} \\ E\_q &= \{(r, s, t) \mid 0 < r < \frac{m}{m+p} \land 0 < s < 1 \land 0 < t < \frac{p}{m+p}\}. \end{aligned} \tag{18}$$

We have convergence domain *r* ⊕*<sup>q</sup> t* < 1, *s* < 1. Consider the function ΦN. We have

$$\begin{array}{l} A\_{q,m+1,n,p} = \frac{\langle a\_1 + m, \beta\_1 + m + p; q \rangle\_1}{\langle \gamma\_1 + m, 1 + m; q \rangle\_1}, \\ A\_{q,m,n+1,p} = \frac{\langle a\_2 + n, \beta\_2 + n; q \rangle\_1}{\langle \gamma\_2 + n + p, 1 + n; q \rangle\_1}, \\ A\_{q,m,n,p+1} = \frac{\langle a\_3 + p, \beta\_1 + m + p; q \rangle\_1}{\langle \gamma\_2 + n + p, 1 + p; q \rangle\_1}. \end{array} \tag{19}$$

We have the following regions

$$X\_q = \{(r, s, t) \mid 0 < s < \frac{n+p}{n} \land 0 < t < \frac{n+p}{p}\}$$

$$Y\_q = \{(r, s, t) \mid 0 < r < \frac{m}{m+p} \land 0 < t < \frac{p}{m+p}\},\tag{20}$$

$$E\_q = \{(r, s, t) \mid 0 < r < \frac{m}{m+p} \land 0 < s < \frac{n+p}{n} \land 0 < t < \frac{n+p}{m+p}\}.$$

We have convergence domain *r* ⊕*<sup>q</sup> t* < 1, *s* < 1.

Consider the function ΦP. We have

$$\begin{aligned} \frac{A\_{q,m+1,n,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_1+m+p, \beta\_1+m+n; q \rangle\_1}{\langle \gamma\_1+m, 1+m; q \rangle\_1},\\ \frac{A\_{q,m,n+1,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_2+n, \beta\_1+m+n; q \rangle\_1}{\langle \gamma\_2+n+p, 1+n; q \rangle\_1},\\ \frac{A\_{q,m,n,p+1}}{A\_{q,m,n,p}} &= \frac{\langle a\_1+m+p, \beta\_2+p; q \rangle\_1}{\langle \gamma\_2+n+p, 1+p; q \rangle\_1}.\end{aligned} \tag{21}$$

We have the following regions

$$\begin{aligned} X\_q &= \{(r, s, t) \mid 0 < s < \frac{n+p}{n} \land 0 < t < \frac{n+p}{p} \} \\ Y\_q &= \{(r, s, t) \mid 0 < r < \frac{m}{m+p} \land 0 < t < \frac{p}{m+p} \} \\ Z\_q &= \{(r, s, t) \mid 0 < r < \frac{m}{m+n} \land 0 < s < \frac{n}{m+n} \} \\ E\_q &= \{(r, s, t) \mid 0 < r < \frac{m^2}{(m+p)(m+n)} \land 0 < s < \frac{n+p}{m+n} \land \\ &\land 0 < t < \frac{n+p}{m+p} \} \end{aligned} \tag{22}$$

We have convergence domain *r* ⊕*<sup>q</sup> t* < 1, *r* ⊕*<sup>q</sup> s* < 1. Consider the function ΦR. We have

$$\begin{split} \frac{A\_{q,m+1,n,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_1+m+p,\beta\_1+m+p;q\rangle\_1}{\langle \gamma\_1+m,1+m;q\rangle\_1},\\ \frac{A\_{q,m,n+1,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_2+n,\beta\_2+n;q\rangle\_1}{\langle \gamma\_2+n+p,1+n;q\rangle\_1},\\ \frac{A\_{q,m,n,p+1}}{A\_{q,m,n,p}} &= \frac{\langle a\_1+m+p,\beta\_1+m+p;q\rangle\_1}{\langle \gamma\_2+n+p,1+p;q\rangle\_1}. \end{split} \tag{23}$$

We have the following regions

$$\begin{aligned} X\_q &= \{(r, s, t) \mid 0 < s < \frac{n+p}{n} \land 0 < t < \frac{n+p}{p}\} \\ Y\_q &= \{(r, s, t) \mid 0 < r < \left(\frac{m}{m+p}\right)^2 \land 0 < t < \left(\frac{p}{m+p}\right)^2\} \\ E\_q &= \{(r, s, t) \mid 0 < r < \left(\frac{m}{m+p}\right)^2 \land 0 < s < \frac{n+p}{n}\land \\ 0 &\land 0 < t < \frac{p(n+p)}{(m+p)^2}\} .\end{aligned} \tag{24}$$

We have convergence domain <sup>√</sup>*<sup>r</sup>* <sup>⊕</sup>*<sup>q</sup>* <sup>√</sup>*<sup>t</sup>* <sup>&</sup>lt; 1, *<sup>s</sup>* <sup>&</sup>lt; 1. The convergence regions for the following two functions are obvious. Consider the function ΦS. We have

$$\begin{split} \frac{A\_{q,m+1,n,p}}{A\_{q,m,n,p}} &= \frac{\langle \alpha\_1 + m\_\prime \beta\_1 + m; q \rangle\_1}{\langle \gamma\_1 + m + n + p, 1 + m; q \rangle\_1}, \\ \frac{A\_{q,m,n+1,p}}{A\_{q,m,n,p}} &= \frac{\langle a\_2 + n + p, \beta\_2 + n; q \rangle\_1}{\langle \gamma\_1 + m + n + p, 1 + n; q \rangle\_1}, \\ \frac{A\_{q,m,n,p+1}}{A\_{q,m,n,p}} &= \frac{\langle a\_2 + n + p, \beta\_3 + p; q \rangle\_1}{\langle \gamma\_1 + m + n + p, 1 + p; q \rangle\_1}. \end{split} \tag{25}$$

Consider the function ΦT. We have

$$\begin{split} \frac{A\_{q,m+1,n,p}}{A\_{q,m,n,p}} &= \frac{\langle \alpha\_1 + m, \beta\_1 + m + p; q \rangle\_1}{\langle \gamma\_1 + m + n + p, 1 + n; q \rangle\_1}, \\\frac{A\_{q,m,n+1,p}}{A\_{q,m,n,p}} &= \frac{\langle \alpha\_2 + n + p, \beta\_2 + n; q \rangle\_1}{\langle \gamma\_1 + m + n + p, 1 + n; q \rangle\_1}, \\\frac{A\_{q,m,n,p+1}}{A\_{q,m,n,p}} &= \frac{\langle \alpha\_2 + n + p, \beta\_1 + m + p; q \rangle\_1}{\langle \gamma\_1 + m + n + p, 1 + p; q \rangle\_1}. \end{split} \tag{26}$$

Consider the function ΦGA . We have

$$\begin{split} \frac{A\_{q,m+1,n,p}}{A\_{q,m,n,p}} &= \frac{\langle \gamma+n+p-m-1, \beta\_1+m+p; q \rangle\_1}{\langle \alpha+n+p-m-1, 1+m; q \rangle\_1}, \\ \frac{A\_{q,m,n+1,p}}{A\_{q,m,n,p}} &= \frac{\langle \alpha+n+p-m, \beta\_2+n; q \rangle\_1}{\langle \gamma+n+p-m, 1+n; q \rangle\_1}, \\ \frac{A\_{q,m,n,p+1}}{A\_{q,m,n,p}} &= \frac{\langle \alpha+n+p-m, \beta\_1+m+p; q \rangle\_1}{\langle \gamma+n+p-m, 1+p; q \rangle\_1}. \end{split} \tag{27}$$

We have the following regions

$$\begin{aligned} Y\_{\emptyset} &= \{(r, s, t) \mid 0 < r < \frac{m}{m+p} \land 0 < t < \frac{p}{m+p}\} \\ E\_{\emptyset} &= \{(r, s, t) \mid 0 < r < \frac{m}{m+p} \land 0 < s < 1 \land 0 < t < \frac{p}{m+p}\}. \end{aligned} \tag{28}$$

We have convergence domain *r* ⊕*<sup>q</sup> t* < 1, *s* < 1. Consider the function ΦGB . We have

$$\begin{split} \frac{A\_{q,m+1,n,p}}{A\_{q,m,n,p}} &= \frac{\langle \gamma+n+p-m-1,\beta\_{1}+m;q\rangle\_{1}}{\langle a+n+p-m-1,1+m;q\rangle\_{1}},\\ \frac{A\_{q,m,n+1,p}}{A\_{q,m,n,p}} &= \frac{\langle a+n+p-m,\beta\_{2}+n;q\rangle\_{1}}{\langle \gamma+n+p-m,1+n;q\rangle\_{1}},\\ \frac{A\_{q,m,n,p+1}}{A\_{q,m,n,p}} &= \frac{\langle a+n+p-m,\beta\_{3}+p;q\rangle\_{1}}{\langle \gamma+n+p-m,1+p;q\rangle\_{1}}.\end{split} \tag{29}$$

The convergence region is obvious.

The convergence region *xy* < *z* for functions Φ<sup>F</sup> and Φ<sup>K</sup> is shown in Figure 1.

**Figure 1.** Convergence region *xy* < *z* for functions Φ<sup>F</sup> and ΦK.

#### **3.** *q***-Integral Representations**

We now turn to *q*-integral expressions of the respective functions. Sometimes we abbreviate the integral ranges by vectors with numbers of elements equal to the numbers of *q*-integrals.

**Theorem 2.** *A triple q-integral representation of* ΦK*. A q-analogue of Dwivedi, Sahai ([11] 4.33). Put*

$$\mathbb{C} \equiv \Gamma\_q \left[ \begin{array}{c} \mathfrak{c}\_1, \mathfrak{c}\_2, \mathfrak{c}\_3 \\ a\_1, b\_1, b\_2, c\_1 - a\_1, c\_2 - b\_2, c\_3 - b\_1 \end{array} \right]. \tag{30}$$

*Then*

$$\begin{split} \Phi\_{K} &= \mathbb{C} \sum\_{m,n,p=0}^{+\infty} \frac{\langle b\_{1}+p;q \rangle\_{m} \langle a\_{2};q \rangle\_{n+p} \mathbf{x}^{m} y^{n} z^{p}}{\langle 1;q \rangle\_{m} \langle 1;q \rangle\_{n} \langle 1;q \rangle\_{p}} \int\_{\vec{0}}^{\vec{1}} u^{a\_{1}+m-1} (qu;q)\_{c\_{1}-a\_{1}-1} \\ & \upsilon^{b\_{2}+n-1} (q\upsilon;q)\_{c\_{2}-b\_{2}-1} \omega^{b\_{1}+p-1} (q\omega;q)\_{c\_{3}-b\_{1}-1} \, d\_{q}(u) \, d\_{q}(v) \, d\_{q}(\omega) . \end{split} \tag{31}$$

**Proof.** The equation numbers in the proof refer to the authors book [12]

$$\begin{aligned} \text{LHS} & \stackrel{\text{by (1.46)}}{=} \sum\_{m,n,p=0}^{+\infty} \frac{\langle a\_2;q \rangle\_{n+p} \langle \overline{b\_1+p};q \rangle\_m x^m y^n z^p}{\langle 1;q \rangle\_m \langle 1;q \rangle\_n \langle 1;q \rangle\_p} \\ \Gamma\_q & \left[ \begin{array}{c} c\_1, c\_2, c\_3, a\_1+m, b\_1+p, b\_2+n \\ a\_1, b\_1, b\_2, c\_1+m, c\_2+n, c\_3+p \end{array} \right] \stackrel{\text{by} 3 \times (7.55)}{=} \text{RHS}. \end{aligned} \tag{32}$$

**Definition 2.** *Assume that <sup>m</sup>* <sup>≡</sup> (*m*1, ... , *mn*), *<sup>m</sup>* <sup>≡</sup> *<sup>m</sup>*<sup>1</sup> <sup>+</sup> ... <sup>+</sup> *mn and <sup>a</sup>* <sup>∈</sup> <sup>R</sup>*. The vector q-multinomial-coefficient* ( *<sup>a</sup> m* ) *<sup>q</sup> [3] is defined by the symmetric expression*

$$\left( \begin{matrix} a \\ \vec{m} \end{matrix} \right)\_q^\* \equiv \frac{\langle -a; q \rangle\_m (-1)^m q^{-\binom{\vec{m}}{2} + am}}{\langle 1; q \rangle\_{m\_1} \langle 1; q \rangle\_{m\_2} \dots \langle 1; q \rangle\_{m\_n}}.\tag{33}$$

The following formula applies for a *q*–deformed hypercube of length 1 in R*n*. Note that formulas (34) and (35) are symmetric in the *xi*.

**Definition 3** ([3])**.** *Assuming that the right hand side converges, and a* <sup>∈</sup> <sup>R</sup>*:*

$$(1 \boxed{}\_{q} q^{a} \mathbf{x}\_{1} \boxed{}\_{q} \dots \boxed{}\_{q} q^{a} \mathbf{x}\_{n})^{-a} \equiv \sum\_{m\_{1}, \dots, m\_{n} = 0}^{\infty} \prod\_{j = 1}^{n} (-\mathbf{x}\_{j})^{m\_{j}} \binom{-a}{\vec{m}}\_{q}^{\*} q^{\binom{\vec{m}}{2} + am} . \tag{34}$$

The following corollary prepares for the next formula.

**Corollary 1.** *A generalization of the q-binomial theorem [3]:*

$$(1\boxplus\_q q^a \mathbf{x}\_1 \boxplus\_q \dots \boxplus\_q q^a \mathbf{x}\_n)^{-a} = \sum\_{\substack{\vec{m}=\vec{0} \\ \vec{m} \gets \vec{0}}}^{\vec{\infty}} \frac{\langle a; q \rangle\_{\mathfrak{m}} \vec{x}^{\vec{m}}}{\langle \vec{1}; q \rangle\_{\vec{m}}}, a \in \mathbb{R}^\star. \tag{35}$$

**Proof.** Use formulas (33) and (34), the terms with factors *q*−( *m* <sup>2</sup> )+*am* cancel each other.

**Theorem 3.** *A double q-integral representation of* Φ<sup>M</sup> *with q-additions. A q-analogue of Saran ([8] 2.13).*

$$\begin{split} \Phi\_{M} &= \Gamma\_{q} \left[ \begin{array}{c} \gamma\_{1}, \gamma\_{2} \\ a\_{1}, a\_{2}, \gamma\_{1} - a\_{1}, \gamma\_{2} - a\_{2} \end{array} \right] \int\_{0}^{1} \int\_{0}^{1} u^{a\_{1} - 1} (qu; q)\_{\gamma\_{1} - a\_{1} - 1} v^{a\_{2} - 1} \\ & (qv; q)\_{\gamma\_{2} - a\_{2} - 1} \frac{1}{(vy; q)\_{\beta\_{2}}} (1 \boxplus\_{q} q^{\beta\_{1}} u x \boxminus\_{q} q^{\beta\_{1}} vz)^{-\beta\_{1}} \, d\_{q}(u) \, d\_{q}(v) . \end{split} \tag{36}$$

**Proof.** The equation numbers in the proof refer to the authors book [12]

LHS = +∞ ∑ *m* =0 *β*2; *q <sup>n</sup>β*1; *q <sup>m</sup>*+*pα*1; *q <sup>m</sup>α*2; *q <sup>n</sup>*+*<sup>p</sup>* 1, *γ*1; *q <sup>m</sup>*1; *q <sup>n</sup>*1; *q <sup>p</sup>γ*2; *q <sup>n</sup>*+*<sup>p</sup> xmynzp* by (1.46) <sup>=</sup> +∞ ∑ *m* =0 *β*2; *q <sup>n</sup>β*1; *q <sup>m</sup>*+*<sup>p</sup>* 1; *q <sup>m</sup>*1; *q <sup>n</sup>*1; *q <sup>p</sup> xmynzp* Γ*<sup>q</sup> γ*1, *γ*2, *α*<sup>1</sup> + *m*, *α*<sup>2</sup> + *n* + *p α*1, *α*2, *γ*<sup>1</sup> + *m*, *γ*<sup>2</sup> + *n* + *p* by (7.55) <sup>=</sup> <sup>Γ</sup>*<sup>q</sup> γ*1, *γ*<sup>2</sup> *α*1, *α*2, *γ*<sup>1</sup> − *α*1, *γ*<sup>2</sup> − *α*<sup>2</sup> <sup>1</sup> 0 <sup>1</sup> 0 *<sup>u</sup>α*1−1(*qu*; *<sup>q</sup>*)*γ*1−*α*1−1*vα*2−1(*qv*; *<sup>q</sup>*)*γ*2−*α*2−<sup>1</sup> +∞ ∑ *m* =0 *β*2; *q <sup>n</sup>β*1; *q <sup>m</sup>*+*<sup>p</sup>* 1; *q <sup>m</sup>*1; *q <sup>n</sup>*1; *q <sup>p</sup>* (*ux*)*m*(*vy*)*n*(*vz*)*<sup>p</sup>* by (7.27),(35) <sup>=</sup> RHS. (37)

**Remark 1.** *Saran ([8] 2.12) gives a similar formula for* Φ<sup>K</sup> *without proof. It is, however, not clear how it is proved.*

All the following vector *q*-integrals have dimension three. We denote *s* ≡ (*s*, *t*, *u*). The short expression to the left always means the definition.

**Theorem 4.** *A q-integral representation of* ΦE*. A q-analogue of ([9] (3.11) p. 22).*

$$\Phi\_{\rm E}(\boldsymbol{a}\_{1},\boldsymbol{a}\_{1},\boldsymbol{\uprho}\_{1},\boldsymbol{\uprho}\_{2},\boldsymbol{\uprho}\_{2};\boldsymbol{\upgamma}\_{1},\boldsymbol{\upgamma}\_{2},\boldsymbol{\upgamma}\_{3}|\boldsymbol{q};\mathbf{x},\boldsymbol{y},\boldsymbol{z})$$

$$\boldsymbol{\upGamma}\_{q}\left[\begin{array}{c}\gamma\_{1},\gamma\_{2},\gamma\_{3}\\\nu\_{1},\nu\_{2},\nu\_{3},\gamma\_{1}-\nu\_{1},\gamma\_{2}-\nu\_{2},\gamma\_{3}-\nu\_{3}\end{array}\right]\int\_{\overline{0}}^{\overline{1}}\overline{s}^{\overline{t}-\overline{1}}(q\vec{s};q)\_{\cdot\overleftarrow{\gamma}-\overline{\nu}-\overline{1}}\tag{38}$$

$$\Phi\_{\rm E}(\boldsymbol{a}\_{1},\boldsymbol{a}\_{1},\boldsymbol{\uprho}\_{1},\boldsymbol{\uprho}\_{2},\boldsymbol{\uprho}\_{2};\boldsymbol{\uprho}\_{1},\boldsymbol{\uprho}\_{2},\boldsymbol{\uprho}\_{3}|\boldsymbol{q};\mathbf{s}\mathbf{x},\boldsymbol{t}\boldsymbol{y},\boldsymbol{u}\boldsymbol{z})\,\boldsymbol{d}\_{\boldsymbol{q}}\overset{\scriptstyle{}}{\operatorname{(}}\mathrm{s}\right).$$

**Proof.** Put

$$\begin{split} D & \equiv \Gamma\_{q} \left[ \begin{array}{c} \gamma\_{1}, \gamma\_{2}, \gamma\_{3} \\ \nu\_{1}, \nu\_{2}, \nu\_{3}, \gamma\_{1} - \nu\_{1}, \gamma\_{2} - \nu\_{2}, \gamma\_{3} - \nu\_{3} \end{array} \right] \\ \sum\_{m,n,p=0}^{+\infty} & \frac{\langle \alpha\_{1}; q \rangle\_{m+n+p} \langle \beta\_{1}; q \rangle\_{m} \langle \beta\_{2}; q \rangle\_{n+p}}{\langle 1, \nu\_{1}; q \rangle\_{m} \langle 1, \nu\_{2}; q \rangle\_{n} \langle 1, \nu\_{3}; q \rangle\_{p}} x^{m} y^{n} z^{p} . \end{split} \tag{39}$$

Then we have (The equation numbers in the proof refer to the authors book [12])

$$\begin{split} &\text{RHS}^{\text{by (\\_{65,0})}} D(1-q)^{3} \sum\_{k,l,j=0}^{+\infty} q^{k(v\_{1}+m)+i(v\_{2}+n)+j(v\_{3}+p)} \\ &\quad \langle 1+k;q \rangle\_{\gamma\_{1}-\nu\_{1}-1} (1+i;q)\_{\gamma\_{2}-\nu\_{2}-1} (1+j;q)\_{\gamma\_{3}-\nu\_{3}-1} \\ &\stackrel{\text{by (6.8.6.10)}{=}} D(1-q)^{3} \sum\_{k,l,j=0}^{+\infty} q^{k(v\_{1}+m)+i(v\_{2}+n)+j(v\_{3}+p)} \\ &\quad \langle \gamma\_{1}-\nu\_{1};q \rangle\_{k} \langle \gamma\_{2}-\nu\_{2};q \rangle\_{j} (\gamma\_{3}-\nu\_{3};q)\_{j} \langle 1,1,1;q \rangle\_{\infty} \\ &\quad \langle \frac{\langle \gamma\_{1}-\nu\_{1};q \rangle\_{k}}{(1;q)\_{k}(1;q)\_{j}(\langle 1;q \rangle\_{j}) \langle \gamma\_{1}-\nu\_{1},\gamma\_{2}-\nu\_{2},\gamma\_{3}-\nu\_{3}\rangle\_{\infty}} \\ &\stackrel{\text{by (2.27)}}{=} D(1-q)^{3} \frac{\langle m+\gamma\_{1},\mathfrak{n}+\gamma\_{2},p+\gamma\_{3},1,1,1;q \rangle\_{\infty}}{\langle \gamma\_{1}+m,\mathfrak{v}\_{2}+\mathfrak{n},\mathfrak{v}\_{3}+p,\gamma\_{1}-\mathfrak{v}\_{1},\gamma\_{2}-\nu\_{2},\gamma\_{3}-\nu\_{3};q \rangle\_{\infty}} \\ &\stackrel{\text{by (1.45),146}}{=} \text{LHS}. \end{split} \tag{2.16}$$

**Theorem 5.** *A q-integral representation of* ΦK*. A q-analogue of ([9] (3.13) p. 23).*

$$\begin{split} \Phi\_{\mathbf{K}} &= \Gamma\_{\eta} \left[ \begin{array}{c} \gamma\_{1}, \gamma\_{2}, \gamma\_{3} \\ \nu\_{1}, \nu\_{2}, \nu\_{3}, \gamma\_{1} - \nu\_{1}, \gamma\_{2} - \nu\_{2}, \gamma\_{3} - \nu\_{3} \end{array} \right] \int\_{\vec{0}}^{\vec{\top}} \vec{s}^{\vec{\top} - \overleftarrow{\top}} (q\vec{s}; q)\_{\vec{\gamma} - \overleftarrow{\nu} - \overleftarrow{\top}} \\ \Phi\_{\mathbf{K}}(a\_{1}, a\_{2}, a\_{2}, \beta\_{1}, \beta\_{2}, \beta\_{1}; \nu\_{1}, \nu\_{2}, \nu\_{3} | q; \mathbf{s}x, ty, uz) \, d\_{\vec{q}}(\mathbf{s}). \end{split} \tag{41}$$

**Proof.** See the proof (40).

**Theorem 6.** *A q-integral representation of* ΦG*. A q-analogue of ([9] (3.12) p. 22).*

$$\begin{split} \Phi\_{\rm G}(a\_{1},a\_{1},\alpha\_{1},\beta\_{1},\beta\_{2},\beta\_{3};\gamma\_{1},\gamma\_{2},\gamma\_{2} | q; \mathbf{x},y,z) \\ = \Gamma\_{\rm q} \left[ \begin{array}{c} \lambda\_{1},\lambda\_{2},\lambda\_{3} \\ \beta\_{1},\beta\_{2},\beta\_{3},\lambda\_{1}-\beta\_{1},\lambda\_{2}-\beta\_{2},\lambda\_{3}-\beta\_{3} \end{array} \right] \int\_{\tilde{0}}^{\tilde{\mathbf{I}}} \tilde{\mathbf{s}}^{\tilde{\mathbf{J}}-\tilde{\mathbf{I}}}(q\tilde{\mathbf{s}};q)\_{\tilde{\lambda}-\tilde{\beta}-\tilde{\mathbf{I}}} \\ \Phi\_{\rm G}(a\_{1},a\_{1},a\_{1},\lambda\_{1},\lambda\_{2},\lambda\_{3};\gamma\_{1},\gamma\_{2},\gamma\_{2} | q; \mathbf{sx},ty,uz) \, d\_{\tilde{\mathbf{q}}}(\mathbf{s}). \end{split} \tag{42}$$

**Proof.** Put

$$\begin{split} D \equiv \Gamma\_{\mathbb{q}} \left[ \begin{array}{c} \lambda\_1, \lambda\_2, \lambda\_3\\ \beta\_1, \beta\_2, \beta\_3, \lambda\_1 - \beta\_1, \lambda\_2 - \beta\_2, \lambda\_3 - \beta\_3 \end{array} \right] \\ \sum\_{m,n,p=0}^{+\infty} \frac{\langle a\_1; q \rangle\_{m+n+p} \langle \lambda\_1; q \rangle\_m \langle \lambda\_2; q \rangle\_n \langle \lambda\_3; q \rangle\_p}{\langle 1, \gamma\_1; q \rangle\_m \langle 1; q \rangle\_n \langle 1; q \rangle\_p \langle \gamma\_2; q \rangle\_{n+p}} x^m y^n z^p. \end{split} \tag{43}$$

Then we have (The equation numbers in the proof refer to the authors book [12])

$$\begin{aligned} \text{RHS} \stackrel{\text{by (6.54)}}{=} & D(1-q)^3 \sum\_{k,i,j=0}^{+\infty} q^{k(\beta\_1+m)+i(\beta\_2+n)+j(\beta\_3+p)}\\ (1+k;q)\_{\lambda\_1-\beta\_1-1}(1+i;q)\_{\lambda\_2-\beta\_2-1}(1+j;q)\_{\lambda\_3-\beta\_3-1} \\ \stackrel{\text{by (6.6.6.10)}{=}} & D(1-q)^3 \sum\_{k,j,j=0}^{+\infty} q^{k(\beta\_1+m)+i(\beta\_2+n)+j(\beta\_3+p)}\\ \frac{\langle\lambda\_1-\beta\_1;q\rangle\langle\lambda\_2-\beta\_2;q\rangle\langle\lambda\_3-\beta\_3;q\rangle\langle1,1,1;q\rangle\langle\infty|}{\langle1;q\rangle\_k\langle1;q\rangle\_l\langle1;q\rangle\_l\langle\lambda\_1-\beta\_1,\lambda\_2-\beta\_2,\lambda\_3-\beta\_3\rangle\langle\infty|}\\ & \stackrel{\text{by (7.27)}}{=} D(1-q)^3 \frac{\langle m+\lambda\_1,n+\lambda\_2,p+\lambda\_3,1,1,1;q\rangle\langle\infty|}{\langle\beta\_1+m,\beta\_2+n,\beta\_3+p,\lambda\_1-\beta\_1,\lambda\_2-\beta\_2,\lambda\_3-\beta\_3;q\rangle\langle\infty|}\\ \text{by (4.51.46)}\end{aligned} \tag{7.28}$$

**Theorem 7.** *A q-integral representation of* ΦN*. A q-analogue of ([9] (3.14) p. 23).*

$$\begin{split} \Phi\_{\mathrm{N}}(\boldsymbol{a}\_{1}, \boldsymbol{a}\_{2}, \boldsymbol{a}\_{3}, \beta\_{1}, \beta\_{2}, \beta\_{1}; \gamma\_{1}, \gamma\_{2}, \gamma\_{2} | \boldsymbol{q}; \mathbf{x}, \boldsymbol{y}, \boldsymbol{z}) \\ = \Gamma\_{\mathrm{q}} \left[ \begin{array}{c} \lambda\_{1}, \lambda\_{2}, \lambda\_{3} \\ \boldsymbol{a}\_{1}, \boldsymbol{a}\_{2}, \boldsymbol{a}\_{3}, \lambda\_{1} - \boldsymbol{a}\_{1}, \lambda\_{2} - \boldsymbol{a}\_{2}, \lambda\_{3} - \boldsymbol{a}\_{3} \end{array} \right] \int\_{\overline{\mathbb{D}}}^{\overline{\mathsf{T}}} \check{\mathsf{s}}^{\overline{\mathsf{T}} - \overline{\mathsf{I}}}(q\check{\mathsf{s}}; q)\_{\overline{\lambda} - \overline{\mathsf{a}} - \overline{\mathsf{I}}} \\ \Phi\_{\mathrm{N}}(\lambda\_{1}, \lambda\_{2}, \lambda\_{3}, \beta\_{1}, \beta\_{2}, \beta\_{1}; \gamma\_{1}, \gamma\_{2}, \gamma\_{2} | q; \mathbf{x}, \boldsymbol{y}, \boldsymbol{u}\_{\boldsymbol{z}} \big) d\_{\boldsymbol{q}}(\boldsymbol{\mathsf{s}}). \end{split} \tag{45}$$

**Proof.** See the proof (44).

**Theorem 8.** *A q-integral representation of* ΦS*. A q-analogue of ([9] (3.15) p. 23).*

$$\Phi\_{\sf S}(\boldsymbol{a}\_{1},\boldsymbol{a}\_{2},\boldsymbol{a}\_{2},\beta\_{1},\beta\_{2},\beta\_{3};\gamma\_{1},\gamma\_{1},\gamma\_{1}|q;\mathbf{x},\boldsymbol{y},\boldsymbol{z})$$

$$\boldsymbol{I} = \Gamma\_{q} \left[ \begin{array}{c} \lambda\_{1},\lambda\_{2},\lambda\_{3} \\ \beta\_{1},\beta\_{2},\beta\_{3},\lambda\_{1}-\beta\_{1},\lambda\_{2}-\beta\_{2},\lambda\_{3}-\beta\_{3} \end{array} \right] \int\_{\tilde{0}}^{\tilde{\mathbf{I}}} \tilde{\mathbf{s}}^{\tilde{\mathbf{J}}-\tilde{\mathbf{I}}}(q\vec{\mathbf{s}};q)\_{\tilde{\lambda}-\tilde{\beta}-\tilde{\mathbf{I}}} \mathbf{I} \tag{46}$$

$$\Phi\_{\sf S}(\boldsymbol{a}\_{1},\boldsymbol{a}\_{2},\boldsymbol{a}\_{2},\lambda\_{1},\lambda\_{2},\lambda\_{3};\gamma\_{1},\gamma\_{1},\gamma\_{1} |q;\mathbf{sx},\boldsymbol{t}y,\boldsymbol{u}\boldsymbol{z}) \,\mathrm{d}\_{\tilde{\mathbf{J}}}(\boldsymbol{s}).$$

**Proof.** See the proof (44).

**Theorem 9.** *A q-integral representation of* ΦF*. A q-analogue of ([9] (3.16) p. 24).*

$$\begin{split} \Phi\_{\rm F}(\alpha\_{1},\alpha\_{1},\alpha\_{1},\beta\_{1},\beta\_{2},\beta\_{1};\gamma\_{1},\gamma\_{2},\gamma\_{2} | q;\mathbf{x},yz,z) \\ = \Gamma\_{\rm q} \left[ \begin{array}{c} \gamma\_{1},\gamma\_{2},\gamma\_{2} \\ \nu\_{1},\nu\_{2},\beta\_{2},\gamma\_{1}-\nu\_{1},\gamma\_{2}-\nu\_{2},\gamma\_{2}-\beta\_{2} \end{array} \right] \\ \end{split} \tag{47}$$
 
$$\begin{split} \int\_{\tilde{0}}^{\tilde{1}} s^{\nu\_{1}-1} t^{\beta\_{2}-1} u^{\nu\_{2}-1} (qs;q)\_{\gamma\_{1}-\nu\_{1}-1} (qt;q)\_{\gamma\_{2}-\beta\_{2}-1} (qu;q)\_{\gamma\_{2}-\nu\_{2}-1} \\ \Phi\_{\rm F}(\alpha\_{1},\alpha\_{1},\alpha\_{1},\beta\_{1},\gamma\_{2},\beta\_{1};\nu\_{1},\nu\_{2},\nu\_{2} | q;\mathbf{s},\mathbf{x},tuyz,uz) \,d\_{q}(\mathbf{s}). \end{split} \tag{48}$$

**Proof.** Put

$$\begin{split} D \equiv \Gamma\_{q} \left[ \begin{array}{c} \gamma\_{1\prime}, \gamma\_{2\prime}, \gamma\_{2} \\ \nu\_{1\prime}, \nu\_{2\prime}, \beta\_{2\prime}, \gamma\_{1} - \nu\_{1\prime}\gamma\_{2} - \nu\_{2\prime}\gamma\_{2} - \beta\_{2} \end{array} \right] \\ \sum\_{m,n,p=0}^{+\infty} \frac{\langle \alpha\_{1}; q \rangle\_{m+n+p} \langle \beta\_{1}; q \rangle\_{m+p} \langle \gamma\_{2}; q \rangle\_{n}}{\langle 1, \nu\_{1}; q \rangle\_{m} \langle 1; q \rangle\_{n} \langle 1; q \rangle\_{p} \langle \nu\_{2}; q \rangle\_{n+p}} x^{m} y^{n} z^{n+p} . \end{split} \tag{48}$$

Then we have (The equation numbers in the proof refer to the authors book [12])

$$\begin{split} &\text{RHS}^{\text{by }\{\underline{6},50\}}D(1-q)^{3}\sum\_{k,j=0}^{+\infty}q^{k(\nu\_{1}+m)+i(\not{p}\_{2}+n)+j(\nu\_{2}+n+p)}\\ &\quad \langle 1+k;q\rangle\_{\gamma\_{1}-\nu\_{1}-1}(1+i;q)\_{\gamma\_{2}-\beta\_{2}-1}(1+j;q)\_{\gamma\_{2}-\nu\_{2}-1}\\ &\stackrel{\text{by }{\langle 6,8,6.10\rangle}}{=}D(1-q)^{3}\sum\_{k,j,j=0}^{+\infty}q^{k(\nu\_{1}+m)+i(\not{p}\_{2}+n)+j(\nu\_{2}+n+p)}\\ &\quad \langle \gamma\_{1}-\nu\_{1};q\rangle\_{k}\langle \gamma\_{2}-\beta\_{2};q\rangle\_{j}\langle \gamma\_{2}-\nu\_{2};q\rangle\_{j}\langle \langle 1,1,1;q\rangle\_{\infty}\\ &\quad \langle 1+\gamma\_{2}\rangle\_{\{1\}}\langle 1;q\rangle\_{\{i\}}\langle \gamma\_{1}-\nu\_{1},\gamma\_{2}-\beta\_{2},\gamma\_{2}-\nu\_{2}\rangle\_{\infty}\\ &\quad \langle \gamma\_{2}\rangle\_{\begin{subarray}{c}\mathcal{V}\subseteq\mathcal{D}\end{subarray}}&\langle m+\gamma\_{1},n+\gamma\_{2},n+p+\gamma\_{2},1,1,1;q\rangle\_{\infty}\\ &\stackrel{\text{by (1.45)},145}{=}\\ &\text{LHS.}\end{split}\tag{49}$$

**Theorem 10.** *A q-integral representation of* ΦM*. A q-analogue of ([9] (3.17) p. 25).*

$$\begin{split} \Phi\_{\mathsf{M}}(\alpha\_{1},\alpha\_{2},\alpha\_{2},\beta\_{1},\beta\_{2},\beta\_{1};\gamma\_{1},\gamma\_{2},\gamma\_{2} | q;\mathsf{x},y;z) \\ = \Gamma\_{q} \left[ \begin{array}{c} \gamma\_{1},\gamma\_{2},\gamma\_{2} \\ \nu\_{1},\nu\_{2},\beta\_{2},\gamma\_{1}-\nu\_{1},\gamma\_{2}-\nu\_{2},\gamma\_{2}-\beta\_{2} \end{array} \right] \\ \end{split} \tag{50}$$
 
$$\begin{split} \int\_{0}^{\overline{1}} \mathbf{s}^{v\_{1}-1} t^{\beta\_{2}-1} u^{v\_{2}-1} (qs;q)\_{\gamma\_{1}-\nu\_{1}-1} (qt;q)\_{\gamma\_{2}-\beta\_{2}-1} (qu;q)\_{\gamma\_{2}-\nu\_{2}-1} \\ \Phi\_{\mathsf{M}}(\alpha\_{1},\alpha\_{2},\alpha\_{2},\beta\_{1},\gamma\_{2},\beta\_{1};\nu\_{1},\nu\_{2},\nu\_{2} | q;\mathsf{x},tu;y,zu) \, d\_{q}(\mathsf{s}). \end{split} \tag{50}$$

**Proof.** Put

$$\begin{split} D \equiv \Gamma\_q \left[ \begin{array}{c} \gamma\_1, \gamma\_2, \gamma\_2 \\ \nu\_1, \nu\_2, \beta\_2, \gamma\_1 - \nu\_1, \gamma\_2 - \nu\_2, \gamma\_2 - \beta\_2 \end{array} \right] \\ \begin{split} \Gamma\_{m,n,p}^{+\infty} \quad \frac{\langle \mathfrak{a}\_1; q \rangle\_m \langle \mathfrak{a}\_2; q \rangle\_{n+p} \langle \beta\_1; q \rangle\_{m+p} \langle \gamma\_2; q \rangle\_n}{\langle 1, \nu\_1; q \rangle\_m \langle 1; q \rangle\_n \langle 1; q \rangle\_p \langle \psi\_2; q \rangle\_{n+p}} \mathfrak{a}^m y^n z^{n+p} . \end{split} \tag{51}$$

Then we have [12]

$$\begin{split} &\text{RHS} \stackrel{\text{by (6.54)}}{=} D(1-q)^{3} \sum\_{k,i,j=0}^{+\infty} q^{k(v\_{1}+m)+i(\beta\_{2}+n)+j(v\_{2}+n+p)} \\ &(1+k;q)\_{\gamma\_{1}-\nu\_{1}-1}(1+i;q)\_{\gamma\_{2}-\beta\_{2}-1}(1+j;q)\_{\gamma\_{2}-\nu\_{2}-1} \\ &\stackrel{\text{by (6.56,10)}}{=} D(1-q)^{3} \sum\_{k,i,j=0}^{+\infty} q^{k(v\_{1}+m)+i(\beta\_{2}+n)+j(v\_{2}+n+p)} \\ &\frac{\langle \gamma\_{1}-\nu\_{1};q\rangle\_{k} \langle \gamma\_{2}-\beta\_{2};q\rangle\_{i} \langle \gamma\_{2}-\nu\_{2};q\rangle\_{j} \langle 1,1,1;q\rangle\_{\infty}}{(1;q)\_{\mathfrak{k}} \langle 1,q\rangle\_{j} \langle 1,q\rangle\_{j} \langle \gamma\_{1}-\nu\_{1},\gamma\_{2}-\beta\_{2},\gamma\_{2}-\nu\_{2}\rangle\_{\infty}} \\ &\stackrel{\text{by (2.27)}}{=} D(1-q)^{3} \frac{\langle m+\gamma\_{1},n+\gamma\_{2},n+p+\gamma\_{2},1,1,1;q\rangle\_{\infty}}{\langle \gamma\_{1}+m,\beta\_{2}+n,\nu\_{2}+n+p,\gamma\_{1}-\nu\_{1},\gamma\_{2}-\nu\_{2},\gamma\_{2}-\beta\_{2};q\rangle\_{\infty}} \\ &\stackrel{\text{by (4.15.14)}}{=} \text{LHS}. \end{split} \tag{2.16}$$

**Theorem 11.** *A q-integral representation of* ΦP*. Almost a q-analogue of ([9] (3.18) p. 25).*

$$\begin{split} \Phi\_{\mathbf{P}}(a\_{1},a\_{2},\mu\_{1},\beta\_{1},\beta\_{1},\beta\_{1},\gamma\_{2},\gamma\_{1},\gamma\_{2},\gamma\_{2} | q;\mathbf{x},\mathbf{z}y,\mathbf{z}) \\ = \Gamma\_{\mathbf{q}} \left[ \begin{array}{c} \gamma\_{1},\gamma\_{2},\gamma\_{2} \\ a\_{2},\nu\_{1},\nu\_{2},\gamma\_{1}-\nu\_{1},\gamma\_{2}-a\_{2},\gamma\_{2}-\nu\_{2} \end{array} \right] \\ \end{split} \tag{53}$$
 
$$\begin{split} \int\_{\widetilde{\mathbf{J}}} \mathbf{s}^{\nu\_{1}-1} t^{\mu\_{2}-1} u^{\nu\_{2}-1} (q\mathbf{s};q)\_{\gamma\_{1}-\nu\_{1}-1} (qt;q)\_{\gamma\_{2}-\mu\_{2}-1} (q\boldsymbol{\mu};q)\_{\gamma\_{2}-\nu\_{2}-1} \\ \Phi\_{\mathbf{P}}(a\_{1},\gamma\_{2},\mu\_{1},\beta\_{1},\beta\_{1},\beta\_{1},\beta\_{2};\nu\_{1},\nu\_{2},\nu\_{2} | q;\mathbf{s}\mathbf{x},tuyz,uz\big) \,d\mathbf{q} \,(\mathbf{s}). \end{split} \tag{53}$$

**Proof.** Put

$$\begin{split} D & \equiv \Gamma\_{\emptyset} \left[ \begin{array}{c} \gamma\_{1\prime}\gamma\_{2\prime}\gamma\_{2} \\ \pi\_{2\prime}\nu\_{1\prime}\nu\_{2\prime}\gamma\_{1} - \nu\_{1\prime}\gamma\_{2} - \alpha\_{2\prime}\gamma\_{2} - \nu\_{2} \end{array} \right] \\ & \sum\_{m,n,p=0}^{+\infty} \frac{\langle a\_{1};q \rangle\_{m+p} \langle \gamma\_{2};q \rangle\_{n} \langle \beta\_{1};q \rangle\_{m+n} \langle \beta\_{2};q \rangle\_{p}}{\langle 1,\nu\_{1};q \rangle\_{m} \langle 1;q \rangle\_{n} \langle 1;q \rangle\_{p} \langle \nu\_{2};q \rangle\_{n+p}} x^{m} y^{n} z^{n+p} . \end{split} \tag{54}$$

Then we have [12]

$$\begin{split} \text{RHS} & \stackrel{\text{by (6.54)}}{=} D(1-q)^{3} \sum\_{k,l,j=0}^{+\infty} q^{k(v\_{1}+m)+i(a\_{2}+n)+j(v\_{2}+n+p)} \\ & (1+k;q)\_{\gamma\_{1}-\nu\_{1}-1}(1+i;q)\_{\gamma\_{2}-a\_{2}-1}(1+j;q)\_{\gamma\_{2}-\nu\_{2}-1} \\ & \stackrel{\text{by (6.5.6)}}{=} D(1-q)^{3} \sum\_{k,l,j=0}^{+\infty} q^{k(v\_{1}+m)+i(a\_{2}+n)+j(v\_{2}+n+p)} \\ & \frac{\langle \gamma\_{1}-\nu\_{1};q\rangle\_{\zeta}\langle \gamma\_{2}-a\_{2};q\rangle\_{\bar{\zeta}}\langle \gamma\_{2}-\nu\_{2};q\rangle\_{\bar{\zeta}}\langle 1,1,1;q\rangle\_{\infty}}{(1;q)\_{\bar{\zeta}}\langle 1;q\rangle\_{\bar{\zeta}}\langle 1;q\rangle\_{\bar{\zeta}}\langle \gamma\_{1}-\nu\_{1},\gamma\_{2}-a\_{2},\gamma\_{2}-\nu\_{2}\rangle\_{\infty}} \\ & \stackrel{\text{by (7.27)}}{=} D(1-q)^{3} \frac{\langle m+\gamma\_{1},n+\gamma\_{2},n+p+\gamma\_{2},1,1,1;q\rangle\_{\infty}}{\langle \gamma\_{1}+m,a\_{2}+n,\nu\_{2}+n+p,\gamma\_{1}-\nu\_{1},\gamma\_{2}-a\_{2},\gamma\_{2}-\nu\_{2};q\rangle\_{\infty}} \\ & \stackrel{\text{by (1.5.14-k)}}{=} \text{LHS}. \end{split} \tag{7.27}$$

**Theorem 12.** *A q-integral representation of* ΦR*. A q-analogue of ([9] (3.19) p. 26).*

$$\begin{split} \Phi\_{\mathsf{R}}(a\_{1},\boldsymbol{\mu}\_{2},\boldsymbol{\mu}\_{1},\beta\_{1},\beta\_{2},\beta\_{1};\gamma\_{1},\gamma\_{2},\gamma\_{2} | q;\mathbf{x},zy,z\rangle \\ = \Gamma\_{q} \left[ \begin{array}{c} \gamma\_{1},\gamma\_{2},\gamma\_{2} \\ \beta\_{2},\nu\_{1},\nu\_{2},\gamma\_{1}-\nu\_{1},\gamma\_{2}-\beta\_{2},\gamma\_{2}-\nu\_{2} \end{array} \right] \\ \end{split} \tag{56}$$
 
$$\begin{split} \frac{\vec{\mathsf{I}}}{\mathsf{d}} \, \mathsf{s}^{\nu\_{1}-1} t^{\beta\_{2}-1} \, \mathsf{u}^{\nu\_{2}-1} (q\mathbf{s};q)\_{\gamma\_{1}-\nu\_{1}-1} (qt;q)\_{\gamma\_{2}-\beta\_{2}-1} (q\mathbf{u};q)\_{\gamma\_{2}-\nu\_{2}-1} \\ \Phi\_{\mathsf{R}}(a\_{1},\nu\_{2},\mu\_{1},\beta\_{1},\gamma\_{2},\beta\_{1};\nu\_{1},\nu\_{2},\nu\_{2} | q;\mathbf{s}x,tuyz,uz\Big) \,d\_{q}\Big(\mathsf{s}\Big). \end{split} \tag{57}$$

**Proof.** See formula (49).

**Theorem 13.** *A q-integral representation of* ΦT*. A q-analogue of ([9] (3.20) p. 27).*

ΦT(*α*1, *α*2, *α*2, *β*1, *β*2, *β*1; *γ*1, *γ*1, *γ*1|*q*; *xz*, *yz*, *z*) = Γ*<sup>q</sup> ξ*, *η*, *γ*<sup>1</sup> *ν*1, *α*1, *β*2, *ξ* − *α*1, *η* − *β*2, *γ*<sup>1</sup> − *ν*<sup>1</sup> 1 0 *s <sup>α</sup>*1−1*t <sup>β</sup>*2−1*uν*1−1(*qs*; *<sup>q</sup>*)*ξ*−*α*1−1(*qt*; *<sup>q</sup>*)*η*−*β*2−1(*qu*; *<sup>q</sup>*)*γ*1−*ν*1−<sup>1</sup> <sup>Φ</sup>T(*ξ*, *<sup>α</sup>*2, *<sup>α</sup>*2, *<sup>β</sup>*1, *<sup>η</sup>*, *<sup>β</sup>*1; *<sup>ν</sup>*1, *<sup>ν</sup>*1, *<sup>ν</sup>*1|*q*;*suxz*, *tuyz*, *uz*) *dq*(*s*). (57)

**Proof.** Put

$$\begin{split} D \equiv \Gamma\_{\mathbb{Q}} \left[ \begin{array}{c} \zeta, \eta, \gamma\_{1} \\ \nu\_{1}, \alpha\_{1}, \beta\_{2}, \xi - \alpha\_{1}, \eta - \beta\_{2}, \gamma\_{1} - \nu\_{1} \end{array} \right] \\ \sum\_{m, n, p = 0}^{+\infty} \frac{\langle \zeta; q \rangle\_{m} \langle \alpha\_{2}; q \rangle\_{n + p} \langle \beta\_{1}; q \rangle\_{m + p} \langle \eta; q \rangle\_{n}}{\langle 1; q \rangle\_{m} \langle 1; q \rangle\_{n} \langle 1; q \rangle\_{p} \langle \nu\_{1}; q \rangle\_{m + n + p}} x^{m} y^{n} z^{m + n + p} . \end{split} \tag{58}$$

Then we have [12]

$$\begin{split} \text{RHS} & \stackrel{\text{by (6.45)}}{=} D(1-q)^{3} \sum\_{k,j=0}^{+\infty} q^{k(a\_{1}+m)+i(\beta\_{2}+n)+j(\upsilon\_{1}+m+n+p)} \\ & (1+k;q)\_{\underline{\zeta}-\mathfrak{a}\_{1}-1}(1+1;q)\_{\upsilon\_{1}-\beta\_{2}-1}(1+j;q)\_{\gamma\_{1}-\upsilon\_{1}-1} \\ & \stackrel{\text{by (6.8.6.10)}{=}}{=} D(1-q)^{3} \sum\_{k,j=0}^{+\infty} q^{k(a\_{1}+m)+i(\beta\_{2}+n)+j(\upsilon\_{1}+m+n+p)} \\ & \frac{(\xi-a\_{1};q)\_{k}(\eta-\beta\_{2};q)\_{j}(\gamma\_{1}-\upsilon\_{1};q)\_{j}(1,1,1;q)\_{\infty}}{(1;q)\_{\mathfrak{k}}(1;q)\_{j}(1,q)\_{j}(\widetilde{\mathfrak{k}}-a\_{1},\eta-\widetilde{\rho}\_{2},\gamma\_{1}-\upsilon\_{1})\_{\infty}} \\ & \stackrel{\text{by (2.7)}}{=} D(1-q)^{3} \frac{\langle m+\widetilde{\zeta},n+\eta,m+n+p+\gamma\_{1},1,1,1;q\rangle\_{\infty}}{\langle a\_{1}+m,\beta\_{2}+n,\upsilon\_{1}+m+n+p,\widetilde{\zeta}-a\_{1},\gamma\_{1}-\upsilon\_{1},\eta-\widetilde{\rho}\_{2};q\rangle\_{\infty}} \\ & \stackrel{\text{by (4.5.14-6)}}{=} \text{LHS}. \end{split} \tag{2.6.8}$$

#### **4. Discussion**

We have successfully combined the convergence condition [13] (*<sup>r</sup>* <sup>⊕</sup>*<sup>q</sup> <sup>t</sup>*)*<sup>n</sup>* <sup>&</sup>lt; 1 with the Horn–Karlsson convergence rules for most of the known triple *q*-hypergeometric functions. The Cartesian equation *r* + *s* + *t* = 1 is thereby replaced by its *q*-analogue *r* ⊕*<sup>q</sup> s* ⊕*<sup>q</sup> t* in the spirit of Rota. The graph for the convergence region *xy*/*z* < 1 could also be of interest for the case *q* = 1. Similarly, the proofs for *q*-Beta integrals also work for the case *q* = 1. These proofs have the same form as in previous and future papers of the author.

#### **5. Conclusions**

In the book [14] more triple hypergeometric functions are discussed. It would be interesting to compute convergence regions for their *q*-analogues. From our convergence theorems it is obvious that the following theorem from ([14], p. 108) can be extended to the *q*-case. The region of convergence for a hypergeometric series is independent of the parameters, exceptional parameter values being excluded. In this way, we plan to write a book on multiple *q*-hypergeometric series.

**Funding:** This research received no external funding

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


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