Transformation formulas and three-term relations for basic hypergeometric series

TL;DR

This paper generalizes Gasper's transformation formulas for basic hypergeometric series and provides explicit coefficients for three-term relations, extending previous results in the field.

Contribution

The paper introduces two new generalized transformation formulas and explicit three-term relation coefficients for {}_{2}phi_{1} series, advancing the understanding of hypergeometric series relations.

Findings

Generalized Gasper's transformation formulas

Explicit coefficients for three-term relations for {}_{2}phi_{1}

Extension of previous results on hypergeometric series

Abstract

We derive two generalizations of Gasper's transformation formula for basic hypergeometric series. Using these generalized formulas, we give explicit expressions for the coefficients of three-term relations for the basic hypergeometric series {}_{2}phi_{1}, which are generalizations of the author's previous results on three-term relations for {}_{2}phi_{1}.

Equations

$$\displaystyle{}_{r+1}\phi_{r}\biggl{(}\genfrac{.}{.}{0.0pt}{}{a,\,b_{1},\dotsc,b_{r}}{c_{1},\dotsc,c_{r}};q,\,x\biggr{)}$$

$$:=\sum _{i=0}^{\infty }\frac{(a{)}_i(b_1{)}_i\cdots (b_r{)}_i}{(q{)}_i(c_1{)}_i\cdots (c_r{)}_i}{x}^i,$$

$$\displaystyle{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{{aq^{k}},\,{bq^{l}}}{cq^{m}};q,\,xq^{n}\biggr{)},\quad{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{{aq^{k^{\prime}}},\,{bq^{l^{\prime}}}}{cq^{m^{\prime}}};q,\,xq^{n^{\prime}}\biggr{)},\quad{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{{a},\,{b}}{c};q,\,x\biggr{)}$$

$$\displaystyle{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{{aq^{k}},\,{bq^{l}}}{cq^{m}};q,\,x\biggr{)}=Q_{0}\cdot{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{{aq},\,{bq}}{cq};q,\,x\biggr{)}+R_{0}\cdot{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{{a},\,{b}}{c};q,\,x\biggr{)}.$$

$$\displaystyle{}_{r+2}\phi_{r+1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{a,\,b,\,c_{1}q^{n_{1}},\dotsc,c_{r}q^{n_{r}}}{bq^{m+1},\,c_{1},\dotsc,c_{r}};q,\,a^{-1}q^{m+1-(n_{1}+\dotsm+n_{r})}\biggr{)}$$

$$=\frac{(q{)}_{\infty }(bq/a{)}_{\infty }(bq{)}_m(c_1/b{)}_{n_1}\cdots (c_r/b{)}_{n_r}}{(q/a{)}_{\infty }(bq{)}_{\infty }(q{)}_m(c_1{)}_{n_1}\cdots (c_r{)}_{n_r}}{b}^{n_1+\cdots +n_r-m}$$

$$\displaystyle\quad\times{}_{r+2}\phi_{r+1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{q^{-m},\,b,\,bq/c_{1},\dotsc,bq/c_{r}}{bq/a,\,bq^{1-n_{1}}/c_{1},\dotsc,bq^{1-n_{r}}/c_{r}};q,\,q\biggr{)}.$$

$$\displaystyle{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{{aq^{k}},\,{bq^{l}}}{cq^{m}};q,\,xq^{n}\biggr{)}=Q\cdot{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{{aq},\,{bq}}{cq};q,\,x\biggr{)}+R\cdot{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{{a},\,{b}}{c};q,\,x\biggr{)}.$$

$$\displaystyle{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{{aq^{k}},\,{bq^{l}}}{cq^{m}};q,\,xq^{n}\biggr{)}=Q^{\prime}\cdot{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{{aq^{k^{\prime}}},\,{bq^{l^{\prime}}}}{cq^{m^{\prime}}};q,\,xq^{n^{\prime}}\biggr{)}+R^{\prime}\cdot{}_{2}\phi_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{{a},\,{b}}{c};q,\,x\biggr{)}$$

$$a,b,c,\frac{a}{b},\frac{c}{a},\frac{c}{b}\notin q^{\mathbb{Z}}\cup \left\{0\right\}.$$

$$Q$$

$$R$$

$$\displaystyle P\biggl{(}\genfrac{.}{.}{0.0pt}{}{{k},\,{l}}{m};{n}\,;\genfrac{.}{.}{0.0pt}{}{{a},\,{b}}{c};q,\,x\biggr{)}$$

$$:=\{\begin{array}{ll}\sum _{j=0}^d\sum _{i=0}^{\max \left\{n,0\right\}}\frac{{\left(q^{-n}\right)}_i}{(q{)}_i}{q}^{ni}\left(A_{j-i+m-\max \left\{m,0\right\}}-B_{j-i-\max \left\{m,0\right\}}\right)x^j,& k+l-m+n\ge 0,\\ \sum _{j=0}^d\sum _{i=0}^{-\min \left\{n,0\right\}}\frac{{\left(q^n\right)}_i}{(q{)}_i}\left({\tilde{A}}_{j-i+m-\max \left\{m,0\right\}}-{\tilde{B}}_{j-i-\max \left\{m,0\right\}}\right)x^j,& k+l-m+n<0,\end{array}$$

$$A_j$$

$$B_j$$

$${\tilde{A}}_j$$

$${\tilde{B}}_j$$

$$\displaystyle Q(k-1,l-1,m-1,n)\bigg{|}_{(a,b,c)\mapsto(aq,bq,cq)}=\frac{(1-aq)(1-bq)x(c-abqx)}{(1-c)(1-cq)}R(k,l,m,n).$$

$$\displaystyle P\biggl{(}\genfrac{.}{.}{0.0pt}{}{{k},\,{l}}{m};{n}\,;\genfrac{.}{.}{0.0pt}{}{{a},\,{b}}{c};q,\,x\biggr{)}$$

$$=\mu \{\begin{array}{ll}\sum _{j=0}^d\sum _{i=0}^{\max \left\{n,0\right\}}\frac{{\left(q^{-n}\right)}_i}{(q{)}_i}{q}^i\left(C_{j-i}-D_{j-i+k-l}\right)x^{d-j},& k+l-m+n\ge 0,\\ \sum _{j=0}^d\sum _{i=0}^{-\min \left\{n,0\right\}}\frac{{\left(q^n\right)}_i}{(q{)}_i}{q}^{(1-n)i}\left({\tilde{C}}_{j-i}-{\tilde{D}}_{j-i+k-l}\right)x^{d-j},& k+l-m+n<0,\end{array}$$

$$C_j$$

$$D_j$$

$${\tilde{C}}_j$$

$${\tilde{D}}_j$$

$${\mu }_1$$

$${\mu }_2$$

$$\mu :=(-1{)}^{k+l-m+M-N-1}{q}^{\left\{k(k-1)+l(l-1)-m(m-1)+M\left(M-1\right)-N\left(N-1\right)\right\}/2}\frac{(q-c)a^k{b}^l}{(b-a)c^m}{\left(\frac{ab}{c}\right)}^M\frac{(c{)}_m}{(a{)}_k(b{)}_l}$$

$$\displaystyle{}_{r+2}\phi_{r+1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{a,\,b,\,c_{1}q^{n_{1}},\dotsc,c_{r}q^{n_{r}}}{bq^{m+1},\,c_{1},\dotsc,c_{r}};q,\,a^{-1}q^{m+1-(n_{1}+\dotsm+n_{r})-s}\biggr{)}$$

$$=\frac{(q{)}_{\infty }(bq/a{)}_{\infty }(bq{)}_m(c_1/b{)}_{n_1}\cdots (c_r/b{)}_{n_r}}{(q/a{)}_{\infty }(bq{)}_{\infty }(q{)}_m(c_1{)}_{n_1}\cdots (c_r{)}_{n_r}}{b}^{n_1+\cdots +n_r-m+s}$$

$$\displaystyle\quad\times{}_{r+2}\phi_{r+1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{q^{-m},\,b,\,bq/c_{1},\dotsc,bq/c_{r}}{bq/a,\,bq^{1-n_{1}}/c_{1},\dotsc,bq^{1-n_{r}}/c_{r}};q,\,q^{1+s}\biggr{)}.$$

$$\sum _{i=0}^s\frac{(q^{-s}{)}_i}{(q{)}_i}{q}^{si}\frac{(q^{1-i}/a{)}_{\infty }(q^{1-n_1}/c_1{)}_{n_1-i}\cdots (q^{1-n_r}/c_r{)}_{n_r-i}}{(q{)}_{\infty }(q^{-m}/b{)}_{m-i+1}}$$

$$\displaystyle\quad\times{}_{r+2}\phi_{r+1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{aq^{i},\,bq^{i},\,c_{1}q^{n_{1}},\dotsc,c_{r}q^{n_{r}}}{bq^{m+1},\,c_{1}q^{i},\dotsc,c_{r}q^{i}};q,\,a^{-1}q^{m+1-(n_{1}+\dotsm+n_{r})+s+(r-2)i}\biggr{)}\allowdisplaybreaks$$

$$=-\sum _{i=0}^s\frac{(q^{-s}{)}_i}{(q{)}_i}{q}^i\frac{(bq/a{)}_{\infty }(b{q}^{1-n_1+i}/c_1{)}_{n_1-i}\cdots (b{q}^{1-n_r+i}/c_r{)}_{n_r-i}}{(b{q}^i{)}_{\infty }(q^{i-m}{)}_{m-i}}{b}^{1-s}$$

$$\displaystyle\quad\times{}_{r+2}\phi_{r+1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{q^{i-m},\,bq^{i},\,bq/c_{1},\dotsc,bq/c_{r}}{bq/a,\,bq^{1-n_{1}+i}/c_{1},\dotsc,bq^{1-n_{r}+i}/c_{r}};q,\,q^{1-s}\biggr{)}.$$

$$\displaystyle\phi_{D}\biggl{(}\genfrac{.}{.}{0.0pt}{}{a;\,b_{1},\dotsc,b_{r}}{c};x_{1},\dotsc,x_{r}\biggr{)}=\frac{(a)_{\infty}(b_{1}x_{1})_{\infty}\!\dotsm(b_{r}x_{r})_{\infty}}{(c)_{\infty}(x_{1})_{\infty}\dotsm(x_{r})_{\infty}}{}_{r+1}\phi_{r}\biggl{(}\genfrac{.}{.}{0.0pt}{}{c/a,\,x_{1},\dotsc,x_{r}}{b_{1}x_{1},\dotsc,b_{r}x_{r}};q,\,a\biggr{)},$$

$$\displaystyle\phi_{D}\biggl{(}\genfrac{.}{.}{0.0pt}{}{a;\,b_{1},\dotsc,b_{r}}{c};x_{1},\dotsc,x_{r}\biggr{)}:=\sum_{i_{1}=0}^{\infty}\dotsm\sum_{i_{r}=0}^{\infty}\frac{(a)_{i_{1}+\dotsm+i_{r}}(b_{1})_{i_{1}}\dotsm(b_{r})_{i_{r}}}{(c)_{i_{1}+\dotsm+i_{r}}(q)_{i_{1}}\dotsm(q)_{i_{r}}}x_{1}^{i_{1}}\dotsm x_{r}^{i_{r}}.$$

$$\displaystyle{}_{r+1}\phi_{r}\biggl{(}\genfrac{.}{.}{0.0pt}{}{q^{-m},\,c_{1}q^{n_{1}},\dotsc,c_{r}q^{n_{r}}}{c_{1},\dotsc,c_{r}};q,\,q\biggr{)}=0.$$

$$(a{)}_i=(a^{-1}{q}^{1-i}{)}_i(-a{)}^i{q}^{i(i-1)/2},(a{)}_{i+j}=(a{)}_i(a{q}^i{)}_j,(a{)}_{i-j}=\frac{(a{)}_i}{(a^{-1}{q}^{1-i}{)}_j}(-a^{-1}{)}^j{q}^{\left\{j(j+1)/2\right\}-ij}.$$

Full text

1. Introduction

In this paper, we derive two generalizations of Gasper’s [7, p. 200, (20)] transformation formula for basic hypergeometric series, and using these generalized formulas, we give explicit expressions for the coefficients of three-term relations for the basic hypergeometric series ${}_{2}\phi_{1}$. These expressions are generalizations of the author’s [11] previous results.

The basic hypergeometric series ${}_{r+1}\phi_{r}$ is defined by

[TABLE]

where $(a)_{i}$ denotes the $q$-shifted factorial defined by $(a)_{i}=(a;q)_{i}:=(a;q)_{\infty}/(aq^{i};q)_{\infty}$ with $(a)_{\infty}=(a;q)_{\infty}:=\prod_{j=0}^{\infty}(1-aq^{j})$. It is assumed that $\lvert q\rvert<1$ and none of the denominator parameters $c_{1},\dotsc,c_{r}$ are $1$ or any negative integer power of $q$.

It is known that for any quadruples of integers $(k,l,m,n)$ and $(k^{\prime},l^{\prime},m^{\prime},n^{\prime})$, the three basic hypergeometric series

[TABLE]

satisfy a linear relation with coefficients that are rational functions of $a,b,c,q$, and $x$. We call such a relation the “three-term relation (for ${}_{2}\phi_{1}$).”

In [11], for any integers $k,l$, and $m$, the three-term relation of the following form is considered:

[TABLE]

The author obtained explicit expressions for $Q_{0}$ and $R_{0}$ using the following transformation formula for basic hypergeometric series obtained by Gasper [7, p. 200, (20)]: If $m,n_{1},\dotsc,n_{r}\in\mathbb{Z}_{\geq 0}$ and \big{\lvert}a^{-1}q^{m+1-(n_{1}+\dotsm+n_{r})}\big{\rvert}<1, then

[TABLE]

In this paper, for any integers $k,l,m$, and $n$, we consider the following three-term relation, which is a generalization of $(\ref{3tr})$:

[TABLE]

Both $(\ref{3tr})$ and $(\ref{gen.3tr})$ are $q$-analogues of the three-term relation for the Gauss hypergeometric series ${}_{2}F_{1}$ considered by Ebisu [5, p. 256, (1.2)].

We derive two generalizations of $(\ref{Gasper.tf})$, and using these generalized formulas, we give explicit expressions for $Q$ and $R$. It is important to note that the more general three-term relation

[TABLE]

can be derived by eliminating ${}_{2}\phi_{1}(aq,bq;cq;q,x)$ from $(\ref{gen.3tr})$ for $(k,l,m,n)$ and $(k^{\prime},l^{\prime},m^{\prime},n^{\prime})$.

Throughout this paper, unless explicitly stated otherwise, we assume that

[TABLE]

1.1. Main results of this paper

Here we present our main results. Note that without loss of generality, it is sufficient to consider $(\ref{gen.3tr})$ for only cases satisfying $k\leq l$, because ${}_{2}\phi_{1}(a,b;c;q,x)$ is symmetric with respect to the exchange of $a$ and $b$.

The following theorem asserts the uniqueness of the pair $(Q,R)$ satisfying $(\ref{gen.3tr})$, and gives explicit expressions for $Q$ and $R$.

Theorem 1**.**

For any integers $k,l,m$, and $n$, there is a unique pair $(Q,R)$ of rational functions of $a,b,c,q$, and $x$ satisfying the three-term relation $(\ref{gen.3tr})$. When $k\leq l$, these functions can be expressed as

[TABLE]

Here, $P$ is the polynomial in $x$ defined by

[TABLE]

where $d:=\max\left\{k+l-m+n,0\right\}+\max\left\{m,0\right\}-\min\left\{n,0\right\}-k-1$ and, for any integer $j$,

[TABLE]

This theorem is a generalization of [11, Lemma 1 and Theorem 2]. (For an expression for $P$