Asian Journal of Mathematics

Volume 25 (2021)

Number 6

Generating functions for Ohno type sums of finite and symmetric multiple zeta-star values

Pages: 871 – 882



Minoru Hirose (Faculty of Mathematics, Kyushu University, Nishi-ku, Fukuoka, Japan; and Institute for Advanced Research, Nagoya University, Chikusa-ku, Nagoya, Japan)

Hideki Murahara (Nakamura Gakuen University Graduate School, Jonan-ku, Fukuoka, Japan; and University of Kitakyushu, Kokuraminami-ku, Kitakyushu, Fukuoka, Japan)

Shingo Saito (Faculty of Arts and Science, Kyushu University, Nishi-ku, Fukuoka, Japan)


Ohno’s relation states that a certain sum, which we call an Ohno type sum, of multiple zeta values remains unchanged if we replace the base index by its dual index. In view of Oyama’s theorem concerning Ohno type sums of finite and symmetric multiple zeta values, Kaneko looked at Ohno type sums of finite and symmetric multiple zeta-star values and made a conjecture on the generating function for a specific index of depth three. In this paper, we confirm this conjecture and further give a formula for arbitrary indices of depth three.


multiple zeta(-star) values, finite multiple zeta(-star) values, symmetric multiple zeta(-star) values, Ohno’s relation, Oyama’s relation

2010 Mathematics Subject Classification

Primary 11M32. Secondary 05A19.

This work was supported by JSPS KAKENHI Grant Numbers JP18J00982, JP18K03243, and JP18K13392.

Received 14 May 2019

Accepted 23 July 2021

Published 24 October 2022