Theta functions and transformations of bilateral basic hypergeometric series

TL;DR

This paper develops new transformation formulas involving theta functions and bilateral basic hypergeometric series, leading to proofs of conjectures and extensions of classical identities.

Contribution

It introduces novel transformation formulas and constructs companion q-series, advancing understanding of asymptotic behaviors and extending Ramanujan and McIntosh identities.

Findings

Proved several conjectures of McIntosh on asymptotic transformations.

Constructed companion q-series with simple closed-form asymptotics.

Extended classical identities of Ramanujan and McIntosh.

Abstract

We establish new transformation formulas involving theta functions and certain bilateral basic hypergeometric series. From these formulas, we construct companion $q$-series for a class of $q$-series such that the asymptotic expansion of their quotient admits a simple closed form. This allows us to prove several conjectures of McIntosh on asymptotic transformations of $q$-series. Moreover, our results extend some identities of Ramanujan and McIntosh.