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

Abstract

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

3 pages