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

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 \begin{equation*} \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 }. \end{equation*}