The united proofs for three $q$-extensions of Dougall's $_2H_2$ summation formula

TL;DR

This paper provides unified proofs for three $q$-extensions of Dougall's $_2H_2$ summation formula using analytic continuation, and discusses related results in the context of basic hypergeometric series.

Contribution

It introduces a unified proof approach for multiple $q$-extensions of Dougall's $_2H_2$ summation formula, expanding understanding of basic hypergeometric identities.

Findings

Unified proofs for three $q$-extensions of Dougall's $_2H_2$ formula

Discussion of related hypergeometric results

Application of analytic continuation method

Abstract

In terms of the analytic continuation method, we give the united proofs for three $q$-extensions of Dougall's $_2H_2$ summation formula. Some related results are also discussed in this paper.

Equations

$$\Gamma (x)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}dt\text{with}\mathfrak{Re}(x)>0.$$

$$(x{)}_n=\frac{\Gamma (x+n)}{\Gamma (x)},$$

$${}_{r}H_{s}\left[\begin{array}[]{cccc}a_{1},&a_{2},&\cdots,&a_{r}\\ b_{1},&b_{2},&\cdots,&b_{s}\end{array}{\!\Big{|}\>z}\right]\>=\>\sum_{k=-\infty}^{\infty}\frac{(a_{1})_{k}(a_{2})_{k}\cdots(a_{r})_{k}}{(b_{1})_{k}(b_{2})_{k}\cdots(b_{s})_{k}}z^{k}.$$

$$\displaystyle{{}_{2}H_{2}}\left[\begin{array}[]{ccc}a,\>b\\ c,\>d\end{array}{\!\Big{|}\>1}\right]=\frac{\Gamma(1-a)\Gamma(1-b)\Gamma(c)\Gamma(d)\Gamma(c+d-a-b-1)}{\Gamma(c-a)\Gamma(c-b)\Gamma(d-a)\Gamma(d-b)},$$

$$(x;q{)}_{\infty }=\prod _{i=0}^{\infty }(1-x{q}^i),(x;q{)}_n=\frac{(x;q{)}_{\infty }}{(x{q}^n;q{)}_{\infty }},$$

$$(x_1,x_2,\cdots ,x_r;q{)}_{\infty }=(x_1;q{)}_{\infty }(x_2;q{)}_{\infty }\cdots (x_r;q{)}_{\infty },$$

$$(x_1,x_2,\cdots ,x_r;q{)}_n=(x_1;q{)}_n(x_2;q{)}_n\cdots (x_r;q{)}_n.$$

$$\displaystyle\quad{{}_{r}\psi_{s}}\left[\begin{array}[]{cccccc}a_{1},&a_{2},&\cdots,&a_{r}\\ b_{1},&b_{2},&\cdots,&b_{s}\end{array}{\!\Big{|}\>q;z}\right]=\sum_{k=-\infty}^{\infty}\frac{(a_{1},a_{2},\cdots,a_{r};q)_{k}}{(b_{1},b_{2},\cdots,b_{s};q)_{k}}\Big{\{}(-1)^{k}q^{\binom{k}{2}}\Big{\}}^{s-r}z^{k}.$$

$$\displaystyle{{}_{1}\psi_{1}}\left[\begin{array}[]{cccccc}a\\ c\end{array}{\!\Big{|}\>q;z}\right]=\frac{(q,c/a,az,q/az;q)_{\infty}}{(c,q/a,z,c/az;q)_{\infty}},$$

$$\displaystyle\quad{{}_{r}\phi_{s-1}}\left[\begin{array}[]{cccccc}a_{1},&a_{2},&\cdots,&a_{r}\\ &b_{2},&\cdots,&b_{s}\end{array}{\!\Big{|}\>q;z}\right]=\sum_{k=0}^{\infty}\frac{(a_{1},a_{2},\cdots,a_{r};q)_{k}}{(q,b_{2},\cdots,b_{s};q)_{k}}\Big{\{}(-1)^{k}q^{\binom{k}{2}}\Big{\}}^{s-r}z^{k}.$$

$$\displaystyle{{}_{2}\psi_{2}}\left[\begin{array}[]{cccccc}a,b\\ c,d\end{array}{\!\Big{|}\>q;z}\right]=\frac{(az,c/b,d/a,qd/abz;q)_{\infty}}{(z,d,q/b,cd/abz;q)_{\infty}}{{}_{2}\psi_{2}}\left[\begin{array}[]{cccccc}a,abz/d\\ c,az\end{array}{\!\Big{|}\>q;\frac{d}{a}}\right],$$

$$\displaystyle{{}_{2}\psi_{2}}\left[\begin{array}[]{cccccc}a,b\\ c,d\end{array}{\!\Big{|}\>q;z}\right]$$

$$\displaystyle{{}_{1}\psi_{1}}\left[\begin{array}[]{cccccc}a\\ c\end{array}{\!\Big{|}\>q;z}\right]=\frac{(q,c/a,az,q/az;q)_{\infty}}{(c,q/a,z,q/z;q)_{\infty}}{{}_{1}\phi_{0}}\left[\begin{array}[]{cccccc}qa/c\\ -\end{array}{\!\Big{|}\>q;\frac{c}{az}}\right].$$

$$\displaystyle{{}_{1}\phi_{0}}\left[\begin{array}[]{cccccc}a\\ -\end{array}{\!\Big{|}\>q;z}\right]=\frac{(az;q)_{\infty}}{(z;q)_{\infty}},$$

$$\displaystyle{{}_{2}\psi_{2}}\left[\begin{array}[]{cccccc}a,b\\ q^{1+m},d\end{array}{\!\Big{|}\>q;z}\right]$$

$${{}_{2}\phi_{1}}\left[\begin{array}[]{cccccc}a,b\\ c\end{array}{\!\Big{|}\>q;z}\right]=\frac{(c/b,bz;q)_{\infty}}{(c,z;q)_{\infty}}{{}_{2}\phi_{1}}\left[\begin{array}[]{cccccc}abz/c,b\\ bz\end{array}{\!\Big{|}\>q;\frac{c}{b}}\right],$$

$$\displaystyle{{}_{2}\psi_{2}}\left[\begin{array}[]{cccccc}a,b\\ q^{1+m},d\end{array}{\!\Big{|}\>q;z}\right]$$

$$\displaystyle\>=\>\frac{(a,b;q)_{-m}z^{-m}}{(q^{1+m},d;q)_{-m}}\frac{(d/a,azq^{-m};q)_{\infty}}{(dq^{-m},z;q)_{\infty}}{{}_{2}\phi_{1}}\left[\begin{array}[]{cccccc}abzq^{-m}/d,aq^{-m}\\ azq^{-m}\end{array}{\!\Big{|}\>q;\frac{d}{a}}\right]$$

$$\displaystyle\>=\>\frac{(a,b;q)_{-m}z^{-m}}{(q^{1+m},d;q)_{-m}}\frac{(d/a,azq^{-m};q)_{\infty}}{(dq^{-m},z;q)_{\infty}}\sum_{k=-m}^{\infty}\frac{(abzq^{-m}/d,aq^{-m};q)_{k+m}}{(q,azq^{-m};q)_{k+m}}\bigg{(}\frac{d}{a}\bigg{)}^{k+m}$$

$$\displaystyle\>=\>\frac{(az,q^{1+m}/b,d/a,qd/abz;q)_{\infty}}{(z,d,q/b,q^{1+m}d/abz;q)_{\infty}}{{}_{2}\psi_{2}}\left[\begin{array}[]{cccccc}a,abz/d\\ q^{1+m},az\end{array}{\!\Big{|}\>q;\frac{d}{a}}\right].$$

$$\displaystyle\sum_{k=0}^{\infty}\frac{(a,b;q)_{k}}{(q^{1+m},d;q)_{k}}z^{k}+\sum_{k=1}^{\infty}\frac{(q/d;q)_{k}\prod_{i=1}^{k}(q^{1+m}-q^{i})}{(q/a,q/b;q)_{k}}\bigg{(}\frac{d}{abz}\bigg{)}^{k}$$

$$=\frac{(az,q^{1+m}/b,d/a,qd/abz;q{)}_{\infty }}{(z,d,q/b,q^{1+m}d/abz;q{)}_{\infty }}$$

$$\displaystyle\>\times\>\bigg{\{}\sum_{k=0}^{\infty}\frac{(a,abz/d;q)_{k}}{(q^{1+m},az;q)_{k}}\bigg{(}\frac{d}{a}\bigg{)}^{k}+\sum_{k=1}^{\infty}\frac{(q/az;q)_{k}\prod_{i=1}^{k}(q^{1+m}-q^{i})}{(q/a,qd/abz;q)_{k}}\bigg{(}\frac{1}{b}\bigg{)}^{k}\bigg{\}}.$$

$$\displaystyle f(c)=\sum_{k=0}^{\infty}\frac{(a,b;q)_{k}}{(c,d;q)_{k}}z^{k}+\sum_{k=1}^{\infty}\frac{(q/d;q)_{k}\prod_{i=1}^{k}(c-q^{i})}{(q/a,q/b;q)_{k}}\bigg{(}\frac{d}{abz}\bigg{)}^{k},$$

$$g(c)=\frac{(az,c/b,d/a,qd/abz;q{)}_{\infty }}{(z,d,q/b,cd/abz;q{)}_{\infty }}$$

$$\displaystyle\qquad\times\>\bigg{\{}\sum_{k=0}^{\infty}\frac{(a,abz/d;q)_{k}}{(c,az;q)_{k}}\bigg{(}\frac{d}{a}\bigg{)}^{k}+\sum_{k=1}^{\infty}\frac{(q/az;q)_{k}\prod_{i=1}^{k}(c-q^{i})}{(q/a,qd/abz;q)_{k}}\bigg{(}\frac{1}{b}\bigg{)}^{k}\bigg{\}}.$$

$$f(c)=g(c)$$

$$\displaystyle{{}_{2}\phi_{1}}\left[\begin{array}[]{cccccc}a,b\\ c\end{array}{\!\Big{|}\>q;z}\right]$$

$$\displaystyle{{}_{2}\psi_{2}}\left[\begin{array}[]{cccccc}a,b\\ q^{1+m},d\end{array}{\!\Big{|}\>q;z}\right]$$

$$=\frac{(a,b;q{)}_{-m}{z}^{-m}}{(q^{1+m},d;q{)}_{-m}}\frac{(b{q}^{-m},d/a,az{q}^{-m},q^{1+m}/az;q{)}_{\infty }}{(d{q}^{-m},b/a,z,q/z;q{)}_{\infty }}$$

$$\displaystyle\>\>\times\>\>{{}_{2}\phi_{1}}\left[\begin{array}[]{cccccc}aq^{-m},qa/d\\ qa/b\end{array}{\!\Big{|}\>q;\frac{q^{1+m}d}{abz}}\right]+idem(a;b)$$

$$\displaystyle\>=\>\frac{(q,b,d/a,az,q/az,q^{1+m}/a;q)_{\infty}}{(q^{1+m},d,b/a,z,q/z,q/a;q)_{\infty}}{{}_{2}\phi_{1}}\left[\begin{array}[]{cccccc}aq^{-m},qa/d\\ qa/b\end{array}{\!\Big{|}\>q;\frac{q^{1+m}d}{abz}}\right]+idem(a;b).$$

$$\displaystyle\sum_{k=0}^{\infty}\frac{(a,b;q)_{k}}{(q^{1+m},d;q)_{k}}z^{k}+\sum_{k=1}^{\infty}\frac{(q/d;q)_{k}\prod_{i=1}^{k}(q^{1+m}-q^{i})}{(q/a,q/b;q)_{k}}\bigg{(}\frac{d}{abz}\bigg{)}^{k}$$

$$\displaystyle\>=\frac{(q,b,d/a,az,q/az,q^{1+m}/a;q)_{\infty}}{(q^{1+m},d,b/a,z,q/z,q/a;q)_{\infty}}\sum_{k=0}^{\infty}\frac{(qa/d;q)_{k}\prod_{i=1}^{k}(q^{1+m}-aq^{i})}{(q,qa/b;q)_{k}}\bigg{(}\frac{d}{abz}\bigg{)}^{k}$$

$$+idem(a;b).$$

$$\displaystyle f(c)=\sum_{k=0}^{\infty}\frac{(a,b;q)_{k}}{(c,d;q)_{k}}z^{k}+\sum_{k=1}^{\infty}\frac{(q/d;q)_{k}\prod_{i=1}^{k}(c-q^{i})}{(q/a,q/b;q)_{k}}\bigg{(}\frac{d}{abz}\bigg{)}^{k},$$

Full text

Email address: weichuanan78@163.com

1. Introduction

For a complex number $x$, define the gamma function by Euler’s integral

[TABLE]

Then the shifted factorial can be expressed as

[TABLE]

where $n$ is an arbitrary integer.

Following Slater [22], define the bilateral hypergeometric series to be

[TABLE]

In 1907, Dougall [12] derived the beautiful identity

[TABLE]

provided $\mathfrak{Re}(c+d-a-b)>1$, according to the contour integral method. Different proofs of it can be found in [2, Section 2.8], [10], [21], [22, Section 6.1], and [23].

Subsequently, define the $q$-shifted factorial by

[TABLE]

where $x$, $q$ are complex numbers satisfying the condition $|q|<1$ and $n$ is an arbitrary integer. For convenience, we shall adopt the two simplified notations:

[TABLE]

Following Gasper and Rahman [13], define the bilateral basic hypergeometric series to be

[TABLE]

Thus Ramanujan’s ${}_{1}\psi_{1}$ summation formula (cf. [13, Appendix II.29]) can be stated as

[TABLE]

provided $|c/a|<|z|<1$. Equation (8) is very important in the theory of special functions. Several beautiful proofs of it can be enjoyed in the papers [3, 4, 5, 9, 18]. The case $b_{1}=q$ of the bilateral basic hypergeometric series is exactly the unilateral basic hypergeometric series

[TABLE]

Then three different $q$-extensions of (4) can be laid out as follows.

Theorem 1**.**

Let $a$, $b$, $c$, $d$ be complex numbers. Then

[TABLE]

where $\max\{|z|,|cd/abz|,|d/a|,|c/b|\}<1$.

Theorem 2**.**

Let $a$, $b$, $c$, $d$ be complex numbers. Then

[TABLE]

where the convergent condition is $\max\{|z|,|cd/abz|\}<1$ and the symbol $idem(a;b)$ after an expression signifies that the front expression is repeated with a and b interchanged.

Theorem 3**.**

Let $a$, $b$, $c$, $d$ be complex numbers. Then

[TABLE]

provided \max\big{\{}|z|,|cd/abz|,|qb/d|\big{\}}<1.

In 1950, Bailey [7] established Theorem 1 by applying the method of comparing coefficients to the product of (8). For the semi-finite form of it, the reader is referred to Chen and Fu [9]. Gasper and Rahman [13, Section 5.4] have shown that there exist two expansions of an ${}_{r}\psi_{r}$ series by means of $r$ ${}_{r}\phi_{r-1}$ series (cf. [13, Equations (5.4.4) and (5.4.5)]). The case $r=2$ of the former is exactly Theorem 2. when $d=b$, it becomes

[TABLE]

Evaluating the ${}_{1}\phi_{0}$ series on the right hand side by $q$-binomial theorem (cf. [13, Appendix II.3]):

[TABLE]

we get Ramanujan’s ${}_{1}\psi_{1}$ summation formula (8). Theorem 3 was deduced by Chens and Gu [8] in accordance with Cauchy’s method. Interestingly, this theorem reduces directly to (8) when $d=a$. Recently, the research of $q$-congruence associated with summation and transformation formulas for the unilateral basic hypergeometric series attracts several mathematicians. Some nice results can be seen in the papers [14, 15, 16, 17].

A property of the analytic function (cf. [20, p.90]; see also [6, 18, 24]), which plays a central role in this paper, can be displayed as the following lemma.

Lemma 4**.**

Let $U$ be a connected open set and $f$, $g$ be analytic on $U$. If $f$ and $g$ agree infinitely often near an interior point of $U$, then we have $f(z)=g(z)$ for all $z\in U$.

The structure of the paper is arranged as follows. By the utilization of Lemma 4, we shall supplies the united proofs of Theorems 1-3 in Section 2. Some related results are also discussed in Section 3.

2. Proofs of Theorems 1-3

§1. Proof of Theorem 1

For a positive integer $m$, we have the relation

[TABLE]

Using (21) and Heine’s transformation formula between two ${}_{2}\phi_{1}$ series (cf. [13, Appendix III.2]):

[TABLE]

we gain

[TABLE]

Split the ${{}_{2}\psi_{2}}$ series on both sides into two parts to achieve

[TABLE]

Define two functions $f(c)$ and $g(c)$ by

[TABLE]

Then (§1) shows that

[TABLE]

for $c=q^{1+m}$. According to Lemma 4, (26) is correct for all $|c|<\min\{1,|abz/d|\}$. By the analytic continuation, the restriction on $c$ can be relaxed. This completes the proof of Theorem 1.

§2. Proof of Theorem 2

By means of (21) and Watson’s transformation formula for three ${}_{2}\phi_{1}$ series (cf. [13, Appendix III.32]):

[TABLE]

we attain

[TABLE]

Split the ${{}_{2}\psi_{2}}$ series on the left hand side into two parts to obtain

[TABLE]

Define two functions $f(c)$ and $g(c)$ by

[TABLE]

Then (§2) gives that

[TABLE]

for $c=q^{1+m}$. In accordance with Lemma 4, (32) is true for all $|c|<1$. Through the analytic continuation, the restriction on $c$ could be relaxed. This finishes the proof of Theorem 2.

§3. Proof of Theorem 3

In terms of (21) and the transformation formula involving three ${}_{2}\phi_{1}$ series (cf. [13, Appendix III.31]):

[TABLE]

we get

[TABLE]

Split the ${{}_{2}\psi_{2}}$-series on the left hand side into two parts to gain

[TABLE]

Define two functions $f(c)$ and $g(c)$ by

[TABLE]

Then (§3) offers that

[TABLE]

for $c=q^{1+m}$. According to Lemma 4, (48) is right for all $|c|<\min\{1,|d/q|,|abz/d|\}$. Via the analytic continuation, the restriction on $c$ can by relaxed. This completes the proof of Theorem 3.

3. Some related discuss

The iteration of Theorem 1 produces another transformation formula between two ${}_{2}\psi_{2}$ series due to Bailey [7]:

[TABLE]

where $\max\{|z|,|cd/abz|\}<1$.

Let $k$ denote the summation index of the ${}_{2}\psi_{2}$ series in Theorem 2. Replace $k$ by $-k$ to achieve

[TABLE]

Employing the substitutions $a\to q/c$, $b\to q/d$, $c\to q/a$, $d\to q/b$, $z\to cd/abz$ in the last equation, we attain the case $r=2$ of [13, Equation (5.4.5)]:

[TABLE]

provided $\max\{|z|,|cd/abz|\}<1$.

Performing the replacements $a\to b$, $b\to a$, $z\to cd/qab$ in Theorem 3, we have

[TABLE]

Calculating the second ${{}_{2}\phi_{1}}$ series on the right hand side by $q$-Gauss summation formula (cf. [13, Appendix II.8]):

[TABLE]

we obtain

[TABLE]

Applying another form of Heine’s transformation (cf. [13, Appendix III.1])

[TABLE]

to the last equation, we recover the following formula due to Chu [11]:

[TABLE]

where the convergent condition is $|cd/qab|<1$.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (No. 11661032).