A $q$-analogue of Euler's formula $ζ(2)=π^2/6$

Zhi-Wei Sun

Published 2018-02-05Version 1

It is well known that $\zeta(2)=\pi^2/6$ as discovered by Euler. In this paper we establish the following $q$-analogue of this celebrated formula: $$\sum_{k=0}^\infty\frac{q^k(1+q^{2k+1})}{(1-q^{2k+1})^2}=\prod_{n=1}^\infty\frac{(1-q^{2n})^4}{(1-q^{2n-1})^4},$$ where $q$ is any complex number with $|q|<1$.