An alternate circular summation formula of theta functions and its applications

TL;DR

This paper introduces a new circular summation formula for theta functions, leading to numerous identities including some from Ramanujan's Notebook, and provides formulas for infinite q-products.

Contribution

The authors present a novel alternate circular summation formula for theta functions, expanding the toolkit for deriving theta identities and related formulas.

Findings

Derived a general theta function summation formula

Recovered several identities from Ramanujan's Notebook

Obtained formulas for (q;q)_^{2n}

Abstract

We prove a general alternate circular summation formula of theta functions, which implies a great deal of theta-function identities. In particular, we recover several identities in Ramanujan's Notebook from this identity. We also obtain two formulaes for $(q;q)_\infty^{2n}$.