On $q$-analogues of some series for $\pi$ and $\pi^2$

TL;DR

This paper develops new $q$-analogues of classical series for $ppa$ and $ppa^2$, extending known formulas into the realm of $q$-series with complex parameter $q$.

Contribution

It introduces novel $q$-analogues of classical series for $ppa$ and $ppa^2$, including a new $q$-analogue of Leibniz's series and two $q$-analogues of a Zeilberger-type series.

Findings

Derived a new $q$-analogue of Leibniz's series for $ppa$.

Established two $q$-analogues of a series for $ppa^2$ involving complex $q$.

Connected classical series for $ppa$ and $ppa^2$ to $q$-series identities.

Abstract

We obtain a new $q$-analogue of the classical Leibniz series $\sum_{k=0}^\infty(-1)^k/(2k+1)=\pi/4$, namely \begin{equation*} \sum_{k=0}^\infty\frac{(-1)^kq^{k(k+3)/2}}{1-q^{2k+1}}=\frac{(q^2;q^2)_{\infty}(q^8;q^8)_{\infty}}{(q;q^2)_{\infty}(q^4;q^8)_{\infty}}, \end{equation*} where $q$ is a complex number with $|q|<1$. We also show that the Zeilberger-type series $\sum_{k=1}^\infty(3k-1)16^k/(k\binom{2k}k)^3=\pi^2/2$ has two $q$-analogues with $|q|<1$, one of which is $$\sum_{n=0}^\infty q^{n(n+1)/2} \frac {1-q^{3n+2}} {1-q} \cdot\frac{(q;q)_n^3 (-q;q)_n}{(q^3;q^2)_{n}^3} = (1-q)^2 \frac{(q^2;q^2)^4_\infty}{(q;q^2)^4_\infty}.$$