Abstract
We provide a q-analogue of Euler’s formula for \(\zeta (2k)\) for \(k\in {\mathbb {Z}}^+\). The result generalizes a recent result of Sun who obtained q-analogues of \(\zeta (2)=\pi ^2/6\) and \(\zeta (4)=\pi ^4/90\). This also extends an earlier result of the present author who obtained a q-analogue of \(\zeta (6)=\pi ^6/945\).
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Acknowlegements
The author is grateful to Krishnaswami Alladi for his constant support, encouragement and stimulating discussions. He sincerely thanks Frank Garvan for several interesting discussions on the problem and providing him with some useful references. He also expresses his appreciation to George Andrews for his support. Finally, he thanks the anonymous referees for their feedback on the manuscript which improved exposition.
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Goswami, A. A q-analogue for Euler’s evaluations of the Riemann zeta function. Res. number theory 5, 3 (2019). https://doi.org/10.1007/s40993-018-0141-y
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DOI: https://doi.org/10.1007/s40993-018-0141-y