q-Analogues of Some Series for Powers of $\pi$

Abstract

We obtain q-analogues of several series for powers of $\pi$. For example, the identity

$$\begin{aligned} \mathop {\sum }\limits _{k=0}^\infty \frac{(-1)^k}{(2k+1)^3}=\frac{\pi ^3}{32} \end{aligned}$$

has the following q-analogue:

$$\begin{aligned} \mathop {\sum }\limits _{k=0}^\infty (-1)^k\frac{q^{2k}(1+q^{2k+1})}{(1-q^{2k+1})^3}=\frac{(q^2;q^4)_{\infty }^2(q^4;q^4)_{\infty }^6}{(q;q^2)_{\infty }^4}, \end{aligned}$$

where q is any complex number with $|q|<1$. We also give q-analogues of four new series for powers of $\pi$ found by the second author.