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

Sumit Kumar Jha

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arXiv · 2111.11857

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

Sumit Kumar Jha

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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.

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Sumit Kumar Jha. 2021-11-21. An identity for $(-q^{5};\,q^{10})_{\infty}$. https://arxiv.org/abs/2111.11857

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