Bilateral \(q\)-series identities and reciprocal formulae
Define \((a)_0:=(a;q)_0:=1,\) \((a)_n:=(a;q)_n:=\prod_{k=0}^{n-1}(1-aq^k),\) \(n\in {\mathbb N},\) \((a)_{\infty}:=(a;q)_{\infty}:=\prod_{k=0}^{\infty}(1-aq^k),\) \(|q|<1.\) In his lost notebook, Ramanujan recorded a beautiful reciprocity theorem, which may be written as follows. Let \(a,q,c\in {\mathbb C},\) \(|q|<1,\) \(c\neq q^{-n}\) with \(n\in {\mathbb N}.\) Then \[ \rho(a,c)-\rho(c,a)=\left(\frac{1}{a}-\frac{1}{c}\right) \frac{(q)_{\infty}(qa/c)_{\infty}(qc/a)_{\infty}}{(qa)_{\infty}(qc)_{\infty}}, \] where \[ \rho(a,c):=\left(1-\frac{1}{c}\right)\sum_{n=0}^{\infty}\frac{(-a/c)^n}{(qa;q)_n}\, q^{\binom{n+1}{2}}. \] The first published proof of this theorem is due to \textit{G. E. Andrews} [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001)]. In the same paper, Andrews generalized Ramanujan's \({}_ 1\psi_1\)-series identity to a four-free-variable reciprocal formula. In the paper under review, the authors show how to obtain similar and more general reciprocal relations exclusively by writing bilateral basic hypergeometric series in terms of two unilateral ones. First, they find several simple reciprocal relations through Ramanujan's \({}_ 1\psi_1\)-series identity. Then, applying Bailey's identity on well-poised \({}_ 6\psi_6\)-series, they generalize Andrew's identity and derive a common extension of Jacobi's triple product identity and the quintuple product identity. Finally, the authors obtain a correct version of the reciprocal formula due to \textit{Z.-Z.~Zhang} [Electron. J. Comb. 12, No. 1, Research paper N3, 3 p. (2005; Zbl 1079.33013)], who proved a false result via the \(q\)-exponential operator method.
Ramanujan's \(_1\psi_1\)-series identity
Bailey's well-poised \(_6\psi_6\)-series identity
scientific article; zbMATH DE number 5758462
scientific article