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CM Evaluations of the Goswami-Sun Series

  • Madeline Locus Dawsey
  • Ken OnoEmail author
Chapter
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)

Abstract

In recent work, Sun constructed two q-series, and he showed that their limits as \(q\rightarrow 1\) give new derivations of the Riemann-zeta values \(\zeta (2)=\pi ^2/6\) and \(\zeta (4)=\pi ^4/90\). Goswami extended these series to an infinite family of q-series, which he analogously used to obtain new derivations of the evaluations of \(\zeta (2k)\in \mathbb {Q}\cdot \pi ^{2k}\) for every positive integer k. Since it is well known that \(\varGamma \left( \frac{1}{2}\right) =\sqrt{\pi }\), it is natural to seek further specializations of these series which involve special values of the \(\varGamma \)-function. Thanks to the theory of complex multiplication, we show that the values of these series at all CM points \(\tau \), where \(q:=e^{2\pi i\tau }\), are algebraic multiples of specific ratios of \(\varGamma \)-values. In particular, classical formulas of Ramanujan allow us to explicitly evaluate these series as algebraic multiples of powers of \(\varGamma \left( \frac{1}{4}\right) ^4/\pi ^3\) when \(q=e^{-\pi }\), \(e^{-2\pi }\).

Notes

Acknowledgements

We thank Krishnaswami Alladi and Ankush Goswami for their beautiful ideas and contributions. We also thank Zhi-Wei Sun for inspiring this work.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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