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Multiple zeta values for classical special functions

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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.

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Notes

  1. This simplification does not happen in the case of terms congruent to \(2 \pmod {3}\) and we obtain the surprising expansion

    $$\begin{aligned} \sum _{n\ge 0}\zeta _{{{\,\mathrm{Ai}\,}}}\left( {\left\{ 2\right\} }^{3n+2}\right) z^{6n}=\frac{\Gamma ^{2}\left( \frac{2}{3}\right) }{\left( 4\pi \right) 3^{\frac{1}{6}}} _{0}F_{3}\left( \begin{array}{c} -\\ \frac{8}{6},\frac{9}{6},\frac{10}{6} \end{array};\frac{z^{6}}{324}\right) . \end{aligned}$$

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

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Authors and Affiliations

  1. University of Maryland, College Park, MD, 20742, USA

    Tanay Wakhare

  2. Tulane University, New Orleans, LA, 70118, USA

    Christophe Vignat

  3. Université Paris Sud, Orsay, France

    Christophe Vignat

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  1. Tanay Wakhare
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  2. Christophe Vignat
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Correspondence to Christophe Vignat.

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This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while Christophe Vignat was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Point Configurations in Geometry, Physics and Computer Science Semester Program, Spring 2018. We thank the anonymous referee for the excellent and straightforward suggestions.

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Wakhare, T., Vignat, C. Multiple zeta values for classical special functions. Ramanujan J 51, 519–551 (2020). https://doi.org/10.1007/s11139-019-00186-5

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