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Multiple zeta values for classical special functions

  • Tanay Wakhare
  • Christophe VignatEmail author
Article
  • 26 Downloads

Abstract

We compute multiple zeta values (MZVs) built from the zeros of various entire functions, usually special functions with physical relevance. In the usual case, MZVs and their linear combinations are evaluated using a morphism between symmetric functions and multiple zeta values. We show that this technique can be extended to the zeros of any entire function, and as an illustration, we explicitly compute some MZVs based on the zeros of Bessel, Airy, and Kummer hypergeometric functions. We highlight several approaches to the theory of MZVs, such as exploiting the orthogonality of various polynomials and fully utilizing the Weierstrass representation of an entire function. On the way, an identity for Bernoulli numbers by Gessel and Viennot is revisited and generalized to Bessel–Bernoulli polynomials, and the classical Euler identity between the Bernoulli numbers and Riemann zeta function at even argument is extended to this same class.

Keywords

Multiple zeta values Bessel, Kummer hypergeometric and Airy functions Zeros of special functions Weierstrass factorization 

Mathematics Subject Classification

Primary 11M32 Secondary 33C10 33C15 

Notes

Acknowledgements

None of the authors have any competing interests in the manuscript.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of MarylandCollege ParkUSA
  2. 2.Tulane UniversityNew OrleansUSA
  3. 3.Université Paris SudOrsayFrance

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