Qing-Hu Hou, Christian Krattenthaler and Zhi-Wei Sun

On q-analogues of some series for π and π2

(12 pages)

Abstract. We obtain two q-analogues of the classical Leibniz series $\sum_{k=0}^\infty(-1)^k/(2k+1)=\pi/4$ , namely

$\sum_{k=0}^\infty\frac{(-q)^k}{1-q^{2k+1}}=\frac{(q^4;q^4)_{\infty}^2}{(q^2;q^4)_{\infty}^2}$

and

\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}},

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

$$\sum_{n=0}^\infty q^{n(n+1)/2} \frac {1-q^{3n+2}} {1-q} \cdot\frac{(q;q)_n^3 (-q;q)_n}{(q^3;q^2)_{n}^3} = (1-q)^2 \frac{(q^2;q^2)^4_\infty}{(q;q^2)^4_\infty}.$$

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