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Fractional hypergeometric zeta functions

Abstract

In this paper we investigate a continuous version of the hypergeometric zeta functions for any positive rational number “a” and demonstrate the analytic continuation. The fractional hypergeometric zeta functions are shown to exhibit many properties analogous to its hypergeometric counter part, including its intimate connection to Bernoulli numbers.

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Acknowledgments

We would like to thank the referee for the helpful comments. We also want to thank Hieu D. Nguyen of the mathematics department at Rowan University for many helpful discussion during the preparation of this paper.

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Correspondence to Abdulkadir Hassen.

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Both authors would like to dedicate this in fond memory of Marvin Knopp. Knop was the most humble and exemplary teacher and mathematician the second author has known.

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Geleta, H.L., Hassen, A. Fractional hypergeometric zeta functions. Ramanujan J 41, 421–436 (2016). https://doi.org/10.1007/s11139-015-9717-5

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Keywords

  • Riemann zeta function
  • Hypergeometric zeta functions
  • Fractional hypergeometric zeta functions
  • Incomplete gamma functions and Confluent hypergeometric function

Mathematics Subject Classification

  • Primary 11M41