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$q$-Analogues of some series for powers of $π$ (1808.04717v2)
Published 13 Aug 2018 in math.CO and math.NT
Abstract: We obtain $q$-analogues of several series for powers of $\pi$. For example, the identity $$\sum_{k=0}\infty\frac{(-1)k}{(2k+1)3}=\frac{\pi3}{32}$$ 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{(q2;q4){\infty}2(q4;q4){\infty}6} {(q;q2)_{\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.