On ${}_5\psi_5$ identities of Bailey

Aritram Dhar

arXiv:2302.14199·math.NT·May 6, 2025

On ${}_5\psi_5$ identities of Bailey

Aritram Dhar

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TL;DR

This paper proves new ${}_5 ext{psi}_5$ identities of Bailey using a ${}_5 ext{phi}_4$ identity, and derives related ${}_3 ext{psi}_3$ formulas through limiting cases, employing techniques from Ismail and Askey.

Contribution

It introduces novel proofs of Bailey's ${}_5 ext{psi}_5$ identities and connects them to ${}_3 ext{psi}_3$ formulas using established summation techniques.

Findings

01

Proved two ${}_5 ext{psi}_5$ summation formulas of Bailey.

02

Derived two ${}_3 ext{psi}_3$ summation formulas as limits.

03

Applied techniques from Ismail and Askey to establish identities.

Abstract

In this paper, we provide proofs of two $_{5} ψ_{5}$ summation formulas of Bailey using a $_{5} ϕ_{4}$ identity of Carlitz. We show that in the limiting case, the two $_{5} ψ_{5}$ identities give rise to two $_{3} ψ_{3}$ summation formulas of Bailey. Finally, we prove the two $_{3} ψ_{3}$ identities using a technique initially used by Ismail to prove Ramanujan's $_{1} ψ_{1}$ summation formula and later by Ismail and Askey to prove Bailey's very-well-poised $_{6} ψ_{6}$ sum.

Equations108

(a)_{n} = (a; q)_{n} := k = 0 \prod n - 1 (1 - a q^{k})

(a)_{n} = (a; q)_{n} := k = 0 \prod n - 1 (1 - a q^{k})

(a)_{\infty} = (a; q)_{\infty} := n \to \infty lim (a; q)_{n} .

(a)_{\infty} = (a; q)_{\infty} := n \to \infty lim (a; q)_{n} .

(a)_{n} := \frac{( a ) _{\infty}}{( a q ^{n} ) _{\infty}} .

(a)_{n} := \frac{( a ) _{\infty}}{( a q ^{n} ) _{\infty}} .

(a_{1}, a_{2}, \dots, a_{k}; q)_{n} := i = 1 \prod k (a_{i}; q)_{n},

(a_{1}, a_{2}, \dots, a_{k}; q)_{n} := i = 1 \prod k (a_{i}; q)_{n},

(a_{1}, a_{2}, \dots, a_{k}; q)_{\infty} := i = 1 \prod k (a_{i}; q)_{\infty} .

(a_{1}, a_{2}, \dots, a_{k}; q)_{\infty} := i = 1 \prod k (a_{i}; q)_{\infty} .

(a; q)_{n + k} = (a; q)_{n} (a q^{n}; q)_{k},

(a; q)_{n + k} = (a; q)_{n} (a q^{n}; q)_{k},

(a; q)_{- n} = \frac{1}{( a q ^{- n} ; q ) _{n}} = \frac{( - q / a ) ^{n}}{( q / a ; q ) _{n}} q^{(2 n)},

(a; q)_{- n} = \frac{1}{( a q ^{- n} ; q ) _{n}} = \frac{( - q / a ) ^{n}}{( q / a ; q ) _{n}} q^{(2 n)},

(a q^{- n}; q)_{k} = \frac{( a ; q ) _{k} ( q / a ; q ) _{n}}{( q ^{1 - k} / a ; q ) _{n}} q^{- nk}, and

(a q^{- n}; q)_{k} = \frac{( a ; q ) _{k} ( q / a ; q ) _{n}}{( q ^{1 - k} / a ; q ) _{n}} q^{- nk}, and

\frac{( a ; q ) _{n - k}}{( b ; q ) _{n - k}} = \frac{( a ; q ) _{n}}{( b ; q ) _{n}} \frac{( q ^{1 - n} / b ; q ) _{k}}{( q ^{1 - n} / a ; q ) _{k}} (\frac{b}{a})^{k} .

\frac{( a ; q ) _{n - k}}{( b ; q ) _{n - k}} = \frac{( a ; q ) _{n}}{( b ; q ) _{n}} \frac{( q ^{1 - n} / b ; q ) _{k}}{( q ^{1 - n} / a ; q ) _{k}} (\frac{b}{a})^{k} .

\begin{multlined}{}_{r}\phi_{r-1}\left[\begin{matrix}a_{1},\ldots,a_{r}\\ \lx@intercol\hfil\begin{matrix}b_{1},\ldots,b_{r-1}\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 1.0pt;q,z\right]:=\sum_{k=0}^{\infty}\dfrac{(a_{1},\ldots,a_{r};q)_{k}}{(q,b_{1},\ldots,b_{r-1};q)_{k}}z^{k},\quad\lvert z\rvert<1,\\ \quad{}_{r}\psi_{r}\left[\begin{matrix}a_{1},\ldots,a_{r}\\ b_{1},\ldots,b_{r}\end{matrix};q,z\right]:=\sum_{k=-\infty}^{\infty}\dfrac{(a_{1},\ldots,a_{r};q)_{k}}{(b_{1},\ldots,b_{r};q)_{k}}z^{k}\quad,\quad\left\lvert\dfrac{b_{1}\ldots b_{r}}{a_{1}\ldots a_{r}}\right\rvert<\lvert z\rvert<1.\end{multlined}{}_{r}\phi_{r-1}\left[\begin{matrix}a_{1},\ldots,a_{r}\\ \lx@intercol\hfil\begin{matrix}b_{1},\ldots,b_{r-1}\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 1.0pt;q,z\right]:=\sum_{k=0}^{\infty}\dfrac{(a_{1},\ldots,a_{r};q)_{k}}{(q,b_{1},\ldots,b_{r-1};q)_{k}}z^{k},\quad\lvert z\rvert<1,\\ \quad{}_{r}\psi_{r}\left[\begin{matrix}a_{1},\ldots,a_{r}\\ b_{1},\ldots,b_{r}\end{matrix};q,z\right]:=\sum_{k=-\infty}^{\infty}\dfrac{(a_{1},\ldots,a_{r};q)_{k}}{(b_{1},\ldots,b_{r};q)_{k}}z^{k}\quad,\quad\left\lvert\dfrac{b_{1}\ldots b_{r}}{a_{1}\ldots a_{r}}\right\rvert<\lvert z\rvert<1.

\begin{multlined}{}_{r}\phi_{r-1}\left[\begin{matrix}a_{1},\ldots,a_{r}\\ \lx@intercol\hfil\begin{matrix}b_{1},\ldots,b_{r-1}\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 1.0pt;q,z\right]:=\sum_{k=0}^{\infty}\dfrac{(a_{1},\ldots,a_{r};q)_{k}}{(q,b_{1},\ldots,b_{r-1};q)_{k}}z^{k},\quad\lvert z\rvert<1,\\ \quad{}_{r}\psi_{r}\left[\begin{matrix}a_{1},\ldots,a_{r}\\ b_{1},\ldots,b_{r}\end{matrix};q,z\right]:=\sum_{k=-\infty}^{\infty}\dfrac{(a_{1},\ldots,a_{r};q)_{k}}{(b_{1},\ldots,b_{r};q)_{k}}z^{k}\quad,\quad\left\lvert\dfrac{b_{1}\ldots b_{r}}{a_{1}\ldots a_{r}}\right\rvert<\lvert z\rvert<1.\end{multlined}{}_{r}\phi_{r-1}\left[\begin{matrix}a_{1},\ldots,a_{r}\\ \lx@intercol\hfil\begin{matrix}b_{1},\ldots,b_{r-1}\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 1.0pt;q,z\right]:=\sum_{k=0}^{\infty}\dfrac{(a_{1},\ldots,a_{r};q)_{k}}{(q,b_{1},\ldots,b_{r-1};q)_{k}}z^{k},\quad\lvert z\rvert<1,\\ \quad{}_{r}\psi_{r}\left[\begin{matrix}a_{1},\ldots,a_{r}\\ b_{1},\ldots,b_{r}\end{matrix};q,z\right]:=\sum_{k=-\infty}^{\infty}\dfrac{(a_{1},\ldots,a_{r};q)_{k}}{(b_{1},\ldots,b_{r};q)_{k}}z^{k}\quad,\quad\left\lvert\dfrac{b_{1}\ldots b_{r}}{a_{1}\ldots a_{r}}\right\rvert<\lvert z\rvert<1.

_{5} ψ_{5} [b, q / b, c, q / c, d, q / d, e, q / e, q^{- n} q^{n + 1}; q, q] = \frac{( q , q / b c , q / b d , q / c d ; q ) _{n}}{( q / b , q / c , q / d , q / b c d ; q ) _{n}}

_{5} ψ_{5} [b, q / b, c, q / c, d, q / d, e, q / e, q^{- n} q^{n + 1}; q, q] = \frac{( q , q / b c , q / b d , q / c d ; q ) _{n}}{( q / b , q / c , q / d , q / b c d ; q ) _{n}}

_{5} ψ_{5} [b, q^{2} / b, c, q^{2} / c, d, q^{2} / d, e, q^{2} / e, q^{- n} q^{n + 2}; q, q] = \frac{( 1 - q ) ( q ^{2} , q ^{2} / b c , q ^{2} / b d , q ^{2} / c d ; q ) _{n}}{( q ^{2} / b , q ^{2} / c , q ^{2} / d , q ^{2} / b c d ; q ) _{n}}

_{5} ψ_{5} [b, q^{2} / b, c, q^{2} / c, d, q^{2} / d, e, q^{2} / e, q^{- n} q^{n + 2}; q, q] = \frac{( 1 - q ) ( q ^{2} , q ^{2} / b c , q ^{2} / b d , q ^{2} / c d ; q ) _{n}}{( q ^{2} / b , q ^{2} / c , q ^{2} / d , q ^{2} / b c d ; q ) _{n}}

_{3} ψ_{3} [b, q / b, c, q / c, d q / d; q, \frac{q}{b c d}] = \frac{( q , q / b c , q / b d , q / c d ; q ) _{\infty}}{( q / b , q / c , q / d , q / b c d ; q ) _{\infty}} .

_{3} ψ_{3} [b, q / b, c, q / c, d q / d; q, \frac{q}{b c d}] = \frac{( q , q / b c , q / b d , q / c d ; q ) _{\infty}}{( q / b , q / c , q / d , q / b c d ; q ) _{\infty}} .

_{3} ψ_{3} [b, q^{2} / b, c, q^{2} / c, d q^{2} / d; q, \frac{q ^{2}}{b c d}] = \frac{( q , q ^{2} / b c , q ^{2} / b d , q ^{2} / c d ; q ) _{\infty}}{( q ^{2} / b , q ^{2} / c , q ^{2} / d , q ^{2} / b c d ; q ) _{\infty}} .

_{3} ψ_{3} [b, q^{2} / b, c, q^{2} / c, d q^{2} / d; q, \frac{q ^{2}}{b c d}] = \frac{( q , q ^{2} / b c , q ^{2} / b d , q ^{2} / c d ; q ) _{\infty}}{( q ^{2} / b , q ^{2} / c , q ^{2} / d , q ^{2} / b c d ; q ) _{\infty}} .

_{1} ψ_{1} [a b; q, z] = \frac{( q , b / a , a z , q / a z ; q ) _{\infty}}{( b , q / a , z , b / a z ; q ) _{\infty}}

_{1} ψ_{1} [a b; q, z] = \frac{( q , b / a , a z , q / a z ; q ) _{\infty}}{( b , q / a , z , b / a z ; q ) _{\infty}}

\begin{multlined}{}_{6}\psi_{6}\left[\begin{matrix}q\sqrt{a},&-q\sqrt{a},&b,&c,&d,&e\\ \sqrt{a},&-\sqrt{a},&aq/b,&aq/c,&aq/d,&aq/e\end{matrix};q,\dfrac{qa^{2}}{bcde}\right]\\ =\dfrac{(aq,aq/bc,aq/bd,aq/be,aq/cd,aq/ce,aq/de,q,q/a;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^{2}/bcde;q)_{\infty}}\end{multlined}{}_{6}\psi_{6}\left[\begin{matrix}q\sqrt{a},&-q\sqrt{a},&b,&c,&d,&e\\ \sqrt{a},&-\sqrt{a},&aq/b,&aq/c,&aq/d,&aq/e\end{matrix};q,\dfrac{qa^{2}}{bcde}\right]\\ =\dfrac{(aq,aq/bc,aq/bd,aq/be,aq/cd,aq/ce,aq/de,q,q/a;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^{2}/bcde;q)_{\infty}}

\begin{multlined}{}_{6}\psi_{6}\left[\begin{matrix}q\sqrt{a},&-q\sqrt{a},&b,&c,&d,&e\\ \sqrt{a},&-\sqrt{a},&aq/b,&aq/c,&aq/d,&aq/e\end{matrix};q,\dfrac{qa^{2}}{bcde}\right]\\ =\dfrac{(aq,aq/bc,aq/bd,aq/be,aq/cd,aq/ce,aq/de,q,q/a;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^{2}/bcde;q)_{\infty}}\end{multlined}{}_{6}\psi_{6}\left[\begin{matrix}q\sqrt{a},&-q\sqrt{a},&b,&c,&d,&e\\ \sqrt{a},&-\sqrt{a},&aq/b,&aq/c,&aq/d,&aq/e\end{matrix};q,\dfrac{qa^{2}}{bcde}\right]\\ =\dfrac{(aq,aq/bc,aq/bd,aq/be,aq/cd,aq/ce,aq/de,q,q/a;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^{2}/bcde;q)_{\infty}}

\begin{multlined}{}_{5}\phi_{4}\left[\begin{matrix}q^{-n},&\quad\quad b,&\quad\quad c,&\quad\quad d,&e\\ \lx@intercol\hfil\begin{matrix}q^{-n+1}/b,&q^{-n+1}/c,&q^{-n+1}/d,&q^{-n+1}/e\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,q\right]\\ =q^{m(1+m-n)}(de)^{-m}\dfrac{(q^{-n})_{2m}(q^{-n+1}/bc,q^{-n+1}/bd,q^{-n+1}/be;q)_{m}}{(q,q^{-n+1}/b,q^{-n+1}/d,q^{-n+1}/e,q^{n-m}c;q)_{m}}(q^{2m-n})_{n-2m}\end{multlined}{}_{5}\phi_{4}\left[\begin{matrix}q^{-n},&\quad\quad b,&\quad\quad c,&\quad\quad d,&e\\ \lx@intercol\hfil\begin{matrix}q^{-n+1}/b,&q^{-n+1}/c,&q^{-n+1}/d,&q^{-n+1}/e\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,q\right]\\ =q^{m(1+m-n)}(de)^{-m}\dfrac{(q^{-n})_{2m}(q^{-n+1}/bc,q^{-n+1}/bd,q^{-n+1}/be;q)_{m}}{(q,q^{-n+1}/b,q^{-n+1}/d,q^{-n+1}/e,q^{n-m}c;q)_{m}}(q^{2m-n})_{n-2m}

\begin{multlined}{}_{5}\phi_{4}\left[\begin{matrix}q^{-n},&\quad\quad b,&\quad\quad c,&\quad\quad d,&e\\ \lx@intercol\hfil\begin{matrix}q^{-n+1}/b,&q^{-n+1}/c,&q^{-n+1}/d,&q^{-n+1}/e\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,q\right]\\ =q^{m(1+m-n)}(de)^{-m}\dfrac{(q^{-n})_{2m}(q^{-n+1}/bc,q^{-n+1}/bd,q^{-n+1}/be;q)_{m}}{(q,q^{-n+1}/b,q^{-n+1}/d,q^{-n+1}/e,q^{n-m}c;q)_{m}}(q^{2m-n})_{n-2m}\end{multlined}{}_{5}\phi_{4}\left[\begin{matrix}q^{-n},&\quad\quad b,&\quad\quad c,&\quad\quad d,&e\\ \lx@intercol\hfil\begin{matrix}q^{-n+1}/b,&q^{-n+1}/c,&q^{-n+1}/d,&q^{-n+1}/e\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,q\right]\\ =q^{m(1+m-n)}(de)^{-m}\dfrac{(q^{-n})_{2m}(q^{-n+1}/bc,q^{-n+1}/bd,q^{-n+1}/be;q)_{m}}{(q,q^{-n+1}/b,q^{-n+1}/d,q^{-n+1}/e,q^{n-m}c;q)_{m}}(q^{2m-n})_{n-2m}

_{3} ϕ_{2} [q^{- 2 m}, \lx@intercol \hfil q^{- 2 m + 1} / a, q^{- 2 m + 1} / b \hfil \lx@intercol a, b; q, \frac{q ^{- m + 2}}{ab}] = \frac{( a , b ; q ) _{m} ( q , ab ; q ) _{2 m}}{( q , ab ; q ) _{m} ( a , b ; q ) _{2 m}} .

_{3} ϕ_{2} [q^{- 2 m}, \lx@intercol \hfil q^{- 2 m + 1} / a, q^{- 2 m + 1} / b \hfil \lx@intercol a, b; q, \frac{q ^{- m + 2}}{ab}] = \frac{( a , b ; q ) _{m} ( q , ab ; q ) _{2 m}}{( q , ab ; q ) _{m} ( a , b ; q ) _{2 m}} .

\begin{multlined}{}_{3}\phi_{2}\left[\begin{matrix}q^{-n},&\quad a,&\quad b\\ \lx@intercol\hfil\begin{matrix}q^{-n+1}/a,&q^{-n+1}/b\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,\dfrac{q^{-n+m+1}z}{ab}\right]\\ =\sum_{2j\leq n}(-1)^{j}\dfrac{(q^{-n})_{2j}(q^{-n+1}/ab)_{j}}{(q,q^{-n+1}/a,q^{-n+1}/b;q)_{j}}q^{-j(j-1)/2+mj}z^{j}(z)_{m-j}(q^{j+m-n}z)_{n-m-j}\end{multlined}{}_{3}\phi_{2}\left[\begin{matrix}q^{-n},&\quad a,&\quad b\\ \lx@intercol\hfil\begin{matrix}q^{-n+1}/a,&q^{-n+1}/b\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,\dfrac{q^{-n+m+1}z}{ab}\right]\\ =\sum_{2j\leq n}(-1)^{j}\dfrac{(q^{-n})_{2j}(q^{-n+1}/ab)_{j}}{(q,q^{-n+1}/a,q^{-n+1}/b;q)_{j}}q^{-j(j-1)/2+mj}z^{j}(z)_{m-j}(q^{j+m-n}z)_{n-m-j}

\begin{multlined}{}_{3}\phi_{2}\left[\begin{matrix}q^{-n},&\quad a,&\quad b\\ \lx@intercol\hfil\begin{matrix}q^{-n+1}/a,&q^{-n+1}/b\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,\dfrac{q^{-n+m+1}z}{ab}\right]\\ =\sum_{2j\leq n}(-1)^{j}\dfrac{(q^{-n})_{2j}(q^{-n+1}/ab)_{j}}{(q,q^{-n+1}/a,q^{-n+1}/b;q)_{j}}q^{-j(j-1)/2+mj}z^{j}(z)_{m-j}(q^{j+m-n}z)_{n-m-j}\end{multlined}{}_{3}\phi_{2}\left[\begin{matrix}q^{-n},&\quad a,&\quad b\\ \lx@intercol\hfil\begin{matrix}q^{-n+1}/a,&q^{-n+1}/b\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,\dfrac{q^{-n+m+1}z}{ab}\right]\\ =\sum_{2j\leq n}(-1)^{j}\dfrac{(q^{-n})_{2j}(q^{-n+1}/ab)_{j}}{(q,q^{-n+1}/a,q^{-n+1}/b;q)_{j}}q^{-j(j-1)/2+mj}z^{j}(z)_{m-j}(q^{j+m-n}z)_{n-m-j}

\begin{multlined}{}_{5}\phi_{4}\left[\begin{matrix}q^{-2m},&\quad bq^{-m},&\quad cq^{-m},&\quad dq^{-m},&\quad eq^{-m}\\ \lx@intercol\hfil\begin{matrix}q^{-m+1}/b,&q^{-m+1}/c,&q^{-m+1}/d,&q^{-m+1}/e\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,q\right]\\ =q^{m^{2}+m}(de)^{-m}\dfrac{(q^{-2m})_{2m}(q/bc,q/bd,q/be;q)_{m}}{(q,q^{-m+1}/b,q^{-m+1}/d,q^{-m+1}/e,c;q)_{m}}\end{multlined}{}_{5}\phi_{4}\left[\begin{matrix}q^{-2m},&\quad bq^{-m},&\quad cq^{-m},&\quad dq^{-m},&\quad eq^{-m}\\ \lx@intercol\hfil\begin{matrix}q^{-m+1}/b,&q^{-m+1}/c,&q^{-m+1}/d,&q^{-m+1}/e\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,q\right]\\ =q^{m^{2}+m}(de)^{-m}\dfrac{(q^{-2m})_{2m}(q/bc,q/bd,q/be;q)_{m}}{(q,q^{-m+1}/b,q^{-m+1}/d,q^{-m+1}/e,c;q)_{m}}

\begin{multlined}{}_{5}\phi_{4}\left[\begin{matrix}q^{-2m},&\quad bq^{-m},&\quad cq^{-m},&\quad dq^{-m},&\quad eq^{-m}\\ \lx@intercol\hfil\begin{matrix}q^{-m+1}/b,&q^{-m+1}/c,&q^{-m+1}/d,&q^{-m+1}/e\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,q\right]\\ =q^{m^{2}+m}(de)^{-m}\dfrac{(q^{-2m})_{2m}(q/bc,q/bd,q/be;q)_{m}}{(q,q^{-m+1}/b,q^{-m+1}/d,q^{-m+1}/e,c;q)_{m}}\end{multlined}{}_{5}\phi_{4}\left[\begin{matrix}q^{-2m},&\quad bq^{-m},&\quad cq^{-m},&\quad dq^{-m},&\quad eq^{-m}\\ \lx@intercol\hfil\begin{matrix}q^{-m+1}/b,&q^{-m+1}/c,&q^{-m+1}/d,&q^{-m+1}/e\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,q\right]\\ =q^{m^{2}+m}(de)^{-m}\dfrac{(q^{-2m})_{2m}(q/bc,q/bd,q/be;q)_{m}}{(q,q^{-m+1}/b,q^{-m+1}/d,q^{-m+1}/e,c;q)_{m}}

_{5} ψ_{5} [b, q / b, c, q / c, d, q / d, e, q / e, q^{- n} q^{n + 1}; q, q]

_{5} ψ_{5} [b, q / b, c, q / c, d, q / d, e, q / e, q^{- n} q^{n + 1}; q, q]

= k = - \infty \sum \infty \frac{( b , c , d , e , q ^{- n} ; q ) _{k}}{( q / b , q / c , q / d , q / e , q ^{n + 1} ; q ) _{k}} q^{k}

= k = - \infty \sum \infty \frac{( b , c , d , e , q ^{- n} ; q ) _{k}}{( q / b , q / c , q / d , q / e , q ^{n + 1} ; q ) _{k}} q^{k}

= k = - n \sum \infty \frac{( b , c , d , e , q ^{- n} ; q ) _{k}}{( q / b , q / c , q / d , q / e , q ^{n + 1} ; q ) _{k}} q^{k} (since 1/ (q^{n + 1})_{k} = 0 for all k < - n)

= k = 0 \sum \infty \frac{( b , c , d , e , q ^{- n} ; q ) _{k - n}}{( q / b , q / c , q / d , q / e , q ^{n + 1} ; q ) _{k - n}} q^{k - n}

= \frac{( b , c , d , e , q ^{- n} ; q ) _{- n} q ^{- n}}{( q / b , q / c , q / d , q / e , q ^{n + 1} ; q ) _{- n}} k = 0 \sum \infty \frac{( q ^{- 2 n} , b q ^{- n} , c q ^{- n} , d q ^{- n} , e q ^{- n} ; q ) _{k}}{( q , q ^{- n + 1} / b , q ^{- n + 1} / c , q ^{- n + 1} / d , q ^{- n + 1} / e ; q ) _{k}} q^{k}

= \frac{( b , c , d , e , q ^{- n} ; q ) _{- n} ( q ^{- 2 n} ) _{2 n} ( q / b c , q / b d , q / b e ; q ) _{n} q ^{n^{2}}}{( q / b , q / c , q / d , q / e , q ^{n + 1} ; q ) _{- n} ( q , q ^{- n + 1} / b , q ^{- n + 1} / d , q ^{- n + 1} / e , c ; q ) _{n} ( d e ) ^{n}}

_{5} ψ_{5} [b, q / b, c, q / c, d, q / d, e, q / e, q^{- n} q^{n + 1}; q, q] = \frac{( q , q / b c , q / b d , q / c d ; q ) _{n}}{( q / b , q / c , q / d , q / b c d ; q ) _{n}}

_{5} ψ_{5} [b, q / b, c, q / c, d, q / d, e, q / e, q^{- n} q^{n + 1}; q, q] = \frac{( q , q / b c , q / b d , q / c d ; q ) _{n}}{( q / b , q / c , q / d , q / b c d ; q ) _{n}}

\begin{multlined}{}_{5}\phi_{4}\left[\begin{matrix}q^{-2m-1},&bq^{-m-1},&cq^{-m-1},&dq^{-m-1},&eq^{-m-1}\\ \lx@intercol\hfil\begin{matrix}q^{-m+1}/b,&q^{-m+1}/c,&q^{-m+1}/d,&q^{-m+1}/e\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,q\right]\\ =(q-1)q^{m^{2}+2m-1}(de)^{-m}\dfrac{(q^{-2m-1})_{2m}(q^{2}/bc,q^{2}/bd,q^{2}/be;q)_{m}}{(q,q^{-m+1}/b,q^{-m+1}/d,q^{-m+1}/e,c;q)_{m}}.\end{multlined}{}_{5}\phi_{4}\left[\begin{matrix}q^{-2m-1},&bq^{-m-1},&cq^{-m-1},&dq^{-m-1},&eq^{-m-1}\\ \lx@intercol\hfil\begin{matrix}q^{-m+1}/b,&q^{-m+1}/c,&q^{-m+1}/d,&q^{-m+1}/e\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,q\right]\\ =(q-1)q^{m^{2}+2m-1}(de)^{-m}\dfrac{(q^{-2m-1})_{2m}(q^{2}/bc,q^{2}/bd,q^{2}/be;q)_{m}}{(q,q^{-m+1}/b,q^{-m+1}/d,q^{-m+1}/e,c;q)_{m}}.

\begin{multlined}{}_{5}\phi_{4}\left[\begin{matrix}q^{-2m-1},&bq^{-m-1},&cq^{-m-1},&dq^{-m-1},&eq^{-m-1}\\ \lx@intercol\hfil\begin{matrix}q^{-m+1}/b,&q^{-m+1}/c,&q^{-m+1}/d,&q^{-m+1}/e\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,q\right]\\ =(q-1)q^{m^{2}+2m-1}(de)^{-m}\dfrac{(q^{-2m-1})_{2m}(q^{2}/bc,q^{2}/bd,q^{2}/be;q)_{m}}{(q,q^{-m+1}/b,q^{-m+1}/d,q^{-m+1}/e,c;q)_{m}}.\end{multlined}{}_{5}\phi_{4}\left[\begin{matrix}q^{-2m-1},&bq^{-m-1},&cq^{-m-1},&dq^{-m-1},&eq^{-m-1}\\ \lx@intercol\hfil\begin{matrix}q^{-m+1}/b,&q^{-m+1}/c,&q^{-m+1}/d,&q^{-m+1}/e\end{matrix}\hfil\lx@intercol\end{matrix}\hskip 2.0pt;q,q\right]\\ =(q-1)q^{m^{2}+2m-1}(de)^{-m}\dfrac{(q^{-2m-1})_{2m}(q^{2}/bc,q^{2}/bd,q^{2}/be;q)_{m}}{(q,q^{-m+1}/b,q^{-m+1}/d,q^{-m+1}/e,c;q)_{m}}.

_{5} ψ_{5} [b, q^{2} / b, c, q^{2} / c, d, q^{2} / d, e, q^{2} / e, q^{- n} q^{n + 2}; q, q]

_{5} ψ_{5} [b, q^{2} / b, c, q^{2} / c, d, q^{2} / d, e, q^{2} / e, q^{- n} q^{n + 2}; q, q]

= k = - \infty \sum \infty \frac{( b , c , d , e , q ^{- n} ; q ) _{k}}{( q ^{2} / b , q ^{2} / c , q ^{2} / d , q ^{2} / e , q ^{n + 2} ; q ) _{k}} q^{k}

= k = - \infty \sum \infty \frac{( b , c , d , e , q ^{- n} ; q ) _{k}}{( q ^{2} / b , q ^{2} / c , q ^{2} / d , q ^{2} / e , q ^{n + 2} ; q ) _{k}} q^{k}

= k = - n - 1 \sum \infty \frac{( b , c , d , e , q ^{- n} ; q ) _{k}}{( q ^{2} / b , q ^{2} / c , q ^{2} / d , q ^{2} / e , q ^{n + 2} ; q ) _{k}} q^{k} (since 1/ (q^{n + 2})_{k} = 0 for all k < - n - 1)

= k = 0 \sum \infty \frac{( b , c , d , e , q ^{- n} ; q ) _{k - n - 1}}{( q ^{2} / b , q ^{2} / c , q ^{2} / d , q ^{2} / e , q ^{n + 2} ; q ) _{k - n - 1}} q^{k - n - 1}

= \frac{( b , c , d , e , q ^{- n} ; q ) _{- n - 1} q ^{- n - 1}}{( q ^{2} / b , q ^{2} / c , q ^{2} / d , q ^{2} / e , q ^{n + 2} ; q ) _{- n - 1}} k = 0 \sum \infty \frac{( q ^{- 2 n - 1} , b q ^{- n - 1} , c q ^{- n - 1} , d q ^{- n - 1} , e q ^{- n - 1} ; q ) _{k}}{( q , q ^{- n + 1} / b , q ^{- n + 1} / c , q ^{- n + 1} / d , q ^{- n + 1} / e ; q ) _{k}} q^{k}

= \frac{( q - 1 ) ( b , c , d , e , q ^{- n} ; q ) _{- n - 1} ( q ^{- 2 n - 1} ) _{2 n} ( q ^{2} / b c , q ^{2} / b d , q ^{2} / b e ; q ) _{n} q ^{n^{2} + n - 2}}{( q ^{2} / b , q ^{2} / c , q ^{2} / d , q ^{2} / e , q ^{n + 2} ; q ) _{- n - 1} ( q , q ^{- n + 1} / b , q ^{- n + 1} / d , q ^{- n + 1} / e , c ; q ) _{n} ( d e ) ^{n}}

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TopicsAdvanced Mathematical Identities · Analytic Number Theory Research · Mathematical Inequalities and Applications

Full text

On ${}_{5}\psi_{5}$ identities of Bailey

Aritram Dhar

Department of Mathematics, University of Florida, Gainesville FL 32611, USA

[email protected]

(Date: August 31, 2023)

Abstract.

In this paper, we provide proofs of two ${}_{5}\psi_{5}$ summation formulas of Bailey using a ${}_{5}\phi_{4}$ identity of Carlitz. We show that in the limiting case, the two ${}_{5}\psi_{5}$ identities give rise to two ${}_{3}\psi_{3}$ summation formulas of Bailey. Finally, we prove the two ${}_{3}\psi_{3}$ identities using a technique initially used by Ismail to prove Ramanujan’s ${}_{1}\psi_{1}$ summation formula and later by Ismail and Askey to prove Bailey’s very-well-poised ${}_{6}\psi_{6}$ sum.

Key words and phrases:

basic hypergeometric series, summation formula, Ismail’s method, Bailey’s ${}_{5}\psi_{5}$ sum, Bailey’s ${}_{3}\psi_{3}$ sum

2020 Mathematics Subject Classification:

33D15, 33D65

1. Introduction

Let $a$ and $q$ be variables and define the conventional $q$ -Pochammer symbol

[TABLE]

for any positive integer $n$ and $(a)_{0}=1$ . For $\lvert q\rvert<1$ , we define

[TABLE]

We define $(a)_{n}$ for all real numbers $n$ by

[TABLE]

For variables $a_{1},a_{2},\ldots,a_{k}$ , we define the shorthand notations

[TABLE]

Next, we require the following formulas from Gasper and Rahman [5, Appendix I]

[TABLE]

We invite the reader to examine Gasper and Rahman’s text [5] for an introduction to basic hypergeometric series, whose notations we follow. For instance, the ${}_{r}\phi_{r-1}$ unilateral and ${}_{r}\psi_{r}$ bilateral basic hypergeometric series with base $q$ and argument $z$ are defined, respectively, by

[TABLE]

Throughout the remainder of this paper, we assume that $\lvert q\rvert<1$ . We now present the statements of the main identities which we prove in this paper.

Theorem 1.1.

(Bailey [2, eq. $3.1$ ]) For any non-negative integer $n$ ,

[TABLE]

where $bcde=q^{n+1}$ .

Theorem 1.2.

(Bailey [2, eq. $3.2$ ]) For any non-negative integer $n$ ,

[TABLE]

where $bcde=q^{n+3}$ .

Theorem 1.3.

(Bailey [2, eq. $2.2$ ])

[TABLE]

Theorem 1.4.

(Bailey [2, eq. $2.3$ ])

[TABLE]

Bailey [2] proved Theorems 1.3 and 1.4 by letting $a\rightarrow 1$ and setting $a=q$ in the ${}_{6}\phi_{5}$ summation formula [5, II. $20$ ] respectively and mentioned that (1.5) and (1.6) follow from Jackson’s $q$ -analogue of Dougall’s theorem [5, II. $22$ ].

Our work is motivated by Ismail’s initial proof [6] of Ramanujan’s ${}_{1}\psi_{1}$ summation formula which can be stated as

[TABLE]

where $\lvert b/a\rvert<\lvert z\rvert<1$ and by Askey and Ismail’s proof [1] of Bailey’s very-well-poised ${}_{6}\psi_{6}$ identity which is

[TABLE]

provided $\lvert qa^{2}/bcde\rvert<1$ .

To prove (1.9) and (1.10), Ismail [6] and Askey and Ismail [1] show that the two sides of (1.9) and (1.10) are analytic functions that agree infinitely often near a point that is an interior point of the domain of analyticity and hence they are identically equal.

To this end, we employ the following $q$ -hypergeometric series identities

Theorem 1.5.

(Carlitz [3, eq. $3.4$ ]) For any non-negative integer $n$ ,

[TABLE]

where $m=\lfloor n/2\rfloor$ and $bcde=q^{1+m-2n}$ .

We note that for $n$ even, Theorem 1.5 is Chu’s [4, p. $279$ ] Corollary $3$ where $\delta=0$ and for $n$ odd, Theorem 1.5 is Chu’s [4, p. $280$ ] Corollary $7$ where $\delta=0$ .

Theorem 1.6.

(Jackson’s terminating q-analogue of Dixon’s sum [5, II. $15$ ]) For any non-negative integer $m$ ,

[TABLE]

Theorem 1.7.

(Carlitz [3, eq. $2.5$ ]) For any non-negative integer $n$ ,

[TABLE]

where $m=\lfloor n/2\rfloor$ .

The paper is organized as follows. In Section 2, we give the proofs of the two ${}_{5}\psi_{5}$ identities (1.5) and (1.6) respectively. In Section 3, we show that the two ${}_{5}\psi_{5}$ identities (1.5) and (1.6) become the two ${}_{3}\psi_{3}$ identities (1.7) and (1.8) respectively when $n\rightarrow\infty$ . Finally, we provide proofs of the two ${}_{3}\psi_{3}$ identities (1.7) and (1.8) in Section 4.

2. Proofs of the two ${}_{5}\psi_{5}$ identities

2.1. Proof of Theorem 1.1

Proof.

Replacing $n$ by $2m$ , $b$ by $bq^{-m}$ , $c$ by $cq^{-m}$ , $d$ by $dq^{-m}$ and $e$ by $eq^{-m}$ in (1.11), we get

[TABLE]

where $bcde=q^{m+1}$ . Now, we have

[TABLE]

where the last equality above follows from (2.1) (after replacing $m$ by $n$ ). Then simplifying the last expression above using (1.1), (1.2) and (1.3) with appropriate substitutions, we get

[TABLE]

where $bcde=q^{n+1}$ for $n\in\mathbb{N}\cup\{0\}$ . This completes the proof of Theorem 1.1. ∎

2.2. Proof of Theorem 1.2

Proof.

Replacing $n$ by $2m+1$ , $b$ by $bq^{-m-1}$ , $c$ by $cq^{-m-1}$ , $d$ by $dq^{-m-1}$ and $e$ by $eq^{-m-1}$ in (1.11), we get

[TABLE]

where $bcde=q^{m+3}$ . Now, we have

[TABLE]

where the last equality above follows from (2.2) (after replacing $m$ by $n$ ). Then simplifying the last expression above using (1.1), (1.2) and (1.3) with appropriate substitutions, we get

[TABLE]

where $bcde=q^{n+3}$ for $n\in\mathbb{N}\cup\{0\}$ . This completes the proof of Theorem 1.2. ∎

3. Two limiting cases

Letting $n\rightarrow\infty$ in (1.5) and simplifying using (1.3) with appropriate substitutions, we get

[TABLE]

which is exactly (1.7).

Similarly, letting $n\rightarrow\infty$ in (1.6) and simplifying using (1.3) with appropriate substitutions, we get

[TABLE]

which is exactly (1.8).

4. Ismail type proofs of the two ${}_{3}\psi_{3}$ identities

In this Section, we derive the the two ${}_{3}\psi_{3}$ identities (1.7) and (1.8) using Ismail’s method [6].

4.1. Proof of Theorem 1.3

Proof.

Replacing $a$ by $bq^{-m}$ and $b$ by $cq^{-m}$ in (1.12), we get

[TABLE]

We now have

[TABLE]

where (4.2) follows using (1.4) with appropriate substitutions and (4.3) follows from (4.1).

Firstly, we note that the series on the left-hand side of (1.7) is an analytic function of $1/d$ provided $\left\lvert q^{2}/bcd\right\rvert<\left\lvert q/bcd\right\rvert<1$ . If we set $1/d=q^{m}$ for any positive integer $m$ in (1.7), we get

[TABLE]

where the last equality above follows from (4.3). Then simplifying the last expression above using (1.1), (1.2) and (1.3) with appropriate substitutions, we get

[TABLE]

Thus, the two sides of (1.7) constitute analytic functions of $1/d$ provided $\left\lvert q^{2}/bcd\right\rvert<\left\lvert q/bcd\right\rvert<1$ where we note that the first of these inequalities always holds simply because $\lvert q\rvert<1$ and the second inequality can be rearranged to give $\lvert 1/d\rvert<\lvert bc/q\rvert$ which is a disk of radius $\lvert bc/q\rvert$ centred about [math]. Thus, both the sides of (1.7) agree on an infinite sequence of points $(q^{m})_{m\in\mathbb{N}}$ which converges to the limit [math] inside the disk $\left\{1/d\in\mathbb{C}:\lvert 1/d\rvert<\lvert bc/q\rvert\right\}$ . Hence, (1.7) is valid in general. This completes the proof of Theorem 1.3. ∎

4.2. Proof of Theorem 1.4

Proof.

Replacing $n$ by $2m+1$ , $z$ by $q^{2}$ , $a$ by $bq^{-m-1}$ and $b$ by $cq^{-m-1}$ in (1.13), we get

[TABLE]

We now have

[TABLE]

where (4.5) follows using (1.4) with appropriate substitutions and (4.6) follows from (4.4).

Firstly, we note that series on the left-hand side of (1.8) is an analytic function of $1/d$ provided $\left\lvert q^{4}/bcd\right\rvert<\left\lvert q^{2}/bcd\right\rvert<1$ . If we set $1/d=q^{m}$ for any positive integer $m$ in (1.8), we get

[TABLE]

where the last equality above follows from (4.6). Then simplifying the last expression above using (1.1), (1.2) and (1.3) with appropriate substitutions, we get

[TABLE]

Thus, the two sides of (1.8) constitute analytic functions of $1/d$ provided $\left\lvert q^{4}/bcd\right\rvert<\left\lvert q^{2}/bcd\right\rvert<1$ where we note that the first of these inequalities always holds simply because $\lvert q\rvert<1$ and the second inequality can be rearranged to give $\lvert 1/d\rvert<\lvert bc/q^{2}\rvert$ which is a disk of radius $\lvert bc/q^{2}\rvert$ centred about [math]. Thus, both the sides of (1.8) agree on an infinite sequence of points $(q^{m})_{m\in\mathbb{N}}$ which converges to the limit [math] inside the disk $\left\{1/d\in\mathbb{C}:\lvert 1/d\rvert<\lvert bc/q^{2}\rvert\right\}$ . Hence, (1.8) is valid in general. This completes the proof of Theorem 1.4. ∎

5. Acknowledgments

The author would like to thank Alexander Berkovich for encouraging him to prove Theorems 1.1, 1.2, 1.3, 1.4 and for his very helpful comments and suggestions. The author would also like to thank George E. Andrews and Jonathan Bradley-Thrush for previewing a preliminary draft of this paper and for their helpful comments. The author would also like to thank the anonymous referee and the editor for their helpful comments.

Bibliography6

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] R. Askey and M. E. H. Ismail, The very well poised ψ 6 6 subscript subscript 𝜓 6 6 {}_{6}\psi_{6} , Proc. Amer. Math. Soc. 77 (1979), no. 2, 218–222.
2[2] W. N. Bailey, On the analogue of Dixon’s theorem for bilateral basic hypergeometric series , Quart. J. Math (Oxford) 2(1) (1950) 194–198.
3[3] L. Carlitz, Some formulas of F. H. Jackson , Monatshefte für Math. 73 (1969), 193–198.
4[4] W. Chu, Abel’s method on summation by parts and bilateral well-poised ψ 5 5 subscript subscript 𝜓 5 5 {}_{5}\psi_{5} -series identities , Port. Math. 66 (3) (2009), 275–302.
5[5] G. Gasper and M. Rahman, Basic Hypergeometric Series , vol. 96, Cambridge University Press, 2004.
6[6] M. Ismail, A simple proof of Ramanujan’s ψ 1 1 subscript subscript 𝜓 1 1 {}_{1}\psi_{1} sum , Proc. Amer. Math. Soc. 63 (1977), no. 1, 185–186.

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On 5ψ5{}_{5}\psi_{5}5​ψ5​ identities of Bailey

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Theorem 1.1**.**

Theorem 1.2**.**

Theorem 1.3**.**

Theorem 1.4**.**

Theorem 1.5**.**

Theorem 1.6**.**

Theorem 1.7**.**

2. Proofs of the two 5ψ5{}_{5}\psi_{5}5​ψ5​ identities

2.1. Proof of Theorem 1.1

Proof.

2.2. Proof of Theorem 1.2

Proof.

3. Two limiting cases

4. Ismail type proofs of the two 3ψ3{}_{3}\psi_{3}3​ψ3​ identities

4.1. Proof of Theorem 1.3

Proof.

4.2. Proof of Theorem 1.4

Proof.

5. Acknowledgments

On ${}_{5}\psi_{5}$ identities of Bailey

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.

Theorem 1.4.

Theorem 1.5.

Theorem 1.6.

Theorem 1.7.

2. Proofs of the two ${}_{5}\psi_{5}$ identities

4. Ismail type proofs of the two ${}_{3}\psi_{3}$ identities