Summation formulae for the bilateral basic hypergeometric series ${}_1\psi_1 ( a; b; q, z )$

TL;DR

This paper derives new summation formulas for the bilateral basic hypergeometric series ${}_1 ext{ extpsi}_1$ using Ramanujan's summation, with implications for mathematical physics and a limit analysis as q approaches 1.

Contribution

It introduces generalized summation formulas for ${}_1 ext{ extpsi}_1$ series based on Ramanujan's work, extending known identities with applications in physics.

Findings

Derived new summation formulas for ${}_1 ext{ extpsi}_1$ series.

Connected the formulas to identities in three-dimensional Abelian mirror symmetry.

Analyzed the limit as q approaches 1, providing insights into the series behavior.

Abstract

We give summation formulae for the bilateral basic hypergeometric series ${}_1\psi_1( a; b; q, z )$ through Ramanujan's summation formula, which are generalizations of nontrivial identities found in the physics of three-dimensional Abelian mirror symmetry on $\mathbf{R}P^2 \times S^1$. We also show the $q \to 1 - 0$ limit of our summation formulae.