An identity for $(-q^{5};\,q^{10})_{\infty}$

TL;DR

This paper proves a new identity involving the infinite product $(-q^{5};q^{10})_{ obreak ext{infinity}}$ using Ramanujan's identities, connecting it to theta functions and sine series.

Contribution

It introduces a novel identity for the infinite product $(-q^{5};q^{10})_{ ext{infinity}}$ derived from Ramanujan's identities, expanding the understanding of q-series and theta functions.

Findings

Established a new product-sum identity involving Ramanujan's identities.

Connected infinite products with theta functions and sine series.

Provided a proof valid for |q|<1.

Abstract

We prove that $$ \prod_{n=0}^{\infty}(1+q^{10n+5}) = \frac{\sum_{n=-\infty}^{\infty}q^{n^{2}}\, \sum_{n=-\infty}^{\infty}(-1)^{n}\, (-q)^{n(3n-1)/2}}{4\, \sum_{n\geq 0}(-q)^{\frac{n(n+1)}{2}}\sin \left\{\frac{(2n+1)3\pi}{10}\right\}\, \sum_{n\geq 0}(-q)^{\frac{n(n+1)}{2}}\sin \left\{\frac{(2n+1)\pi}{10}\right\}} \qquad |q|<1 $$ using identities due to Ramanujan.