Hidden \(q\)-analogues of Ramanujan-like \(\pi\)-series
The authors take the famous series representations for \(1/\pi\) by Ramanujan, and show that the \(q\)-analogue of some of these are actually easily deducible by known identities scattered in the literature. For instance, the classical \(q\)-series equality \[ \begin{aligned} &\sum_{k=0}^{\infty} \frac{1-q^{2 k} a}{1-a}\left[\begin{array}{c} a, b, c, d \\ q, q a / b, q a / c, q a / d \end{array} \mid q\right]_{k}\left(\frac{q a}{b c d}\right)^{k} \\ &=\left[\begin{array}{c} q a, q a / b c, q a / b d, q a / c d \\ q a / b, q a / c, q a / d, q a / b c d \end{array} \mid q\right]_{\infty} \end{aligned} \] implies that \[ \sum_{k=0}^{\infty}(-1)^{k} \frac{1-q^{4 k+1}}{1-q} \frac{\left(q ; q^{2}\right)_{k}^{3}}{\left(q^{2} ; q^{2}\right)_{k}^{3}} q^{k^{2}}=\Gamma_{q^{2}}\left[\begin{array}{cc} 1,1 \\ \frac{1}{2}, \frac{3}{2} \end{array}\right]. \] This, in turn, is the \(q\)-analogue of \[ \sum_{k=0}^{\infty}(-1)^{k}\left\{\frac{\left(\frac{1}{2}\right)_{k}}{k !}\right\}^{3}(1+4 k)=\frac{2}{\pi}. \] Many similar identities and analogues, based on quadratic, cubic, quartic, and quintic series, are mentioned in the paper.
scientific article; zbMATH DE number 7344949
scientific article