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Generating function of multiple polylog of Hurwitz type

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  21 November 2022

Kentaro Ihara ,
Yusuke Kusunoki ,
Yayoi Nakamura Open the ORCID record for Yayoi Nakamura [Opens in a new window]  and
Hitomi Saeki
Kentaro Ihara*
Affiliation:
Faculty of Science and Engineering, Kindai University, 3-4-1 Kowakae, Higashi-Osaka, Japan
Yusuke Kusunoki
Affiliation:
EX Corporation, 3-19-3 Toyosaki, Osaka, Japan e-mail: kusunokiyusuke1130@gmail.com
Yayoi Nakamura
Affiliation:
Faculty of Science and Engineering, Kindai University, 3-4-1 Kowakae, Higashi-Osaka, Japan e-mail: yayoi@math.kindai.ac.jp
Hitomi Saeki
Affiliation:
Uenomiya Taishi Senior High School, Minamikawachi, Osaka, Japan e-mail: shu.apple2k@gmail.com
*
e-mail: k-ihara@math.kindai.ac.jp
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Abstract

We introduce interpolated multiple Hurwitz polylogs and interpolated multiple Hurwitz zeta values. In addition, we discuss the generating functions for the sum of the polylogs/zeta values of fixed weight, depth, and all heights. The functions are expressed in terms of generalized hypergeometric functions. Compared with the pioneering results of Ohno and Zagier on the generating function, our setup generalizes the results in three directions, namely, at general heights, with a t-interpolation, and as a Hurwitz type. As an application, by fixing the Hurwitz parameter to rational numbers, the generating functions for multiple zeta values with level are given.

Keywords

Multiple zeta valuesHurwitz zeta functionhypergeometric functions

MSC classification

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values 11M35: Hurwitz and Lerch zeta functions
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$ 33C20: Generalized hypergeometric series, $_pF_q$
Type
Article
Information
Canadian Journal of Mathematics , Volume 76 , Issue 1 , February 2024 , pp. 1 - 17
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The first author is supported by JSPS KAKENHI Grant Number 18K03260.

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