$q$-Analogues of some series for powers of $\pi$

TL;DR

This paper develops $q$-analogues of classical series for powers of $$, including new formulas, providing a bridge between series for $$ and $q$-series with potential applications in combinatorics and number theory.

Contribution

The paper introduces novel $q$-analogues of known series for powers of $$, extending classical identities to the $q$-series setting.

Findings

Derived $q$-analogue of a series for $^3$

Established $q$-analogues for four new series for powers of $$

Provided explicit formulas involving $q$-series and infinite products

Abstract

We obtain $q$-analogues of several series for powers of $\pi$. For example, the identity $$\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^3}=\frac{\pi^3}{32}$$ has the following $q$-analogue: \begin{equation*} \sum_{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{equation*} 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.