$q$-Analogues of some series for powers of $π$ (1808.04717v2)

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.