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A q-analogue for Euler’s evaluations of the Riemann zeta function

  • Ankush GoswamiEmail author
Research
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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 \(\zeta (6)=\pi ^6/945\).

Keywords

Riemann zeta function Stirling numbers of second kind Triangular numbers Upper half plane 

Mathematics Subject Classification

11N25 11N37 11N60 

Notes

Author's contributions

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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Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

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