Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T14:19:49.118Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A WEIGHTED SUM OF MULTIPLE $\mathbf{{T}}$-VALUES OF FIXED WEIGHT AND DEPTH

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

Published online by Cambridge University Press:  19 March 2021

YOSHIHIRO TAKEYAMA Open the ORCID record for YOSHIHIRO TAKEYAMA [Opens in a new window]
YOSHIHIRO TAKEYAMA*
Affiliation:
Department of Mathematics, Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
*
e-mail: takeyama@math.tsukuba.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

The multiple T-value, which is a variant of the multiple zeta value of level two, was introduced by Kaneko and Tsumura [‘Zeta functions connecting multiple zeta values and poly-Bernoulli numbers’, in: Various Aspects of Multiple Zeta Functions, Advanced Studies in Pure Mathematics, 84 (Mathematical Society of Japan, Tokyo, 2020), 181–204]. We show that the generating function of a weighted sum of multiple T-values of fixed weight and depth is given in terms of the multiple T-values of depth one by solving a differential equation of Heun type.

Keywords

Heun’s equationhypergeometric functionmultiple T-value

MSC classification

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Secondary: 33C90: Applications
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 3 , December 2021 , pp. 398 - 405
Check for updates
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was partially supported by JSPS KAKENHI grant number 18K03233.

References

Berger, S., Chandra, A., Jain, J., Xu, D., Xu, C. and Zhao, J., ‘Weighted sums of Euler sums and other variants of multiple zeta values’, Preprint, 2021, arXiv:2011.02393.Google Scholar
Ishkhanyan, A. and Suominen, K. A., ‘New solutions of Heun’s general equation’, J. Phys. A 36(5) (2003), L81–L85.10.1088/0305-4470/36/5/101CrossRefGoogle Scholar
Kaneko, M. and Tsumura, H., ‘On a variant of multiple zeta values of level two’, Preprint, 2019, arXiv:1903.03747.10.21099/tkbjm/20204402213CrossRefGoogle Scholar
Kaneko, M. and Tsumura, H., ‘Zeta functions connecting multiple zeta values and poly-Bernoulli numbers’, in: Various Aspects of Multiple Zeta Functions—in Honour of Professor Kohji Matsumoto’s 60th Birthday, Advanced Studies in Pure Mathematics, 84 (Mathematical Society of Japan, Tokyo, 2020), 181–204.10.2969/aspm/08410181CrossRefGoogle Scholar
Ohno, Y. and Zagier, D., ‘Multiple zeta values of fixed weight, depth, and height’, Indag. Math. (N.S.) 12(4) (2001), 483–487.10.1016/S0019-3577(01)80037-9CrossRefGoogle Scholar
Sasaki, Y., ‘On generalized poly-Bernoulli numbers and related $L$ -functions’, J. Number Theory 132(1) (2012), 156–170.CrossRefGoogle Scholar