A note on Kawashima functions
Shuji Yamamoto
Publications Mathématiques de Besançon no. 1  (2019), p. 151-163

This note is a survey of results on the function F k (z) introduced by G. Kawashima, and its applications to the study of multiple zeta values. We stress the viewpoint that the Kawashima function is a generalization of the digamma function ψ(z), and explain how various formulas for ψ(z) are generalized. We also discuss briefly the relationship of the results on the Kawashima functions with the recent work on Kawashima’s MZV relation by M. Kaneko and the author.

L’objet de cette note est de faire une revue des résultats sur la fonction F k (z) définie par G. Kawashima et des applications à l’étude des valeurs de fonctions zêtas multiples. Nous mettons l’accent sur le fait que cette fonction de Kawashima est une généralisation de la fonction digamma ψ(z) et nous expliquons comment des formules valables pour ψ(z) se généralisent. Nous survolons également les liens entre les résultats sur les fonctions de G. Kawashima avec les travaux récents des relations MZV de Kawashima de M. Kaneko et de l’auteur.

Published online : 2019-10-15
Classification:  11M32,  33B15
Keywords: Kawashima functions, Digamma function, Polygamma functions, Multiple zeta values
     author = {Shuji Yamamoto},
     title = {A note on Kawashima functions},
     journal = {Publications Math\'ematiques de Besan\c con},
     publisher = {Presses universitaires de Franche-Comt\'e},
     number = {1},
     year = {2019},
     pages = {151-163},
     language = {en},
     url = {https://pmb.centre-mersenne.org/item/PMB_2019___1_151_0}
Yamamoto, Shuji. A note on Kawashima functions. Publications Mathématiques de Besançon, no. 1 (2019), pp. 151-163. pmb.centre-mersenne.org/item/PMB_2019___1_151_0/

[1] Michael E. Hoffman Quasi-symmetric functions and mod p multiple harmonic sums, Kyushu J. Math., Tome 69 (2015) no. 2, pp. 345-366 | Zbl 1382.11066

[2] Masanobu Kaneko; Shuji Yamamoto A new integral-series identity of multiple zeta values and regularizations, Sel. Math., New Ser., Tome 24 (2018) no. 3, pp. 2499-2521 | Zbl 06904447

[3] Gaku Kawashima A class of relations among multiple zeta values, J. Number Theory, Tome 129 (2009) no. 4, pp. 755-788 | Zbl 1220.11103

[4] Gaku Kawashima Multiple series expressions for the Newton series which interpolate finite multiple harmonic sums (2009) (https://arxiv.org/abs/0905.0243)

[5] Yoshihiro Takeyama Quadratic relations for a q-analogue of multiple zeta values, Ramanujan J., Tome 27 (2012) no. 1, pp. 15-28 | Zbl 1305.05021

[6] Tatsushi Tanaka On the quasi-derivation relation for multiple zeta values, J. Number Theory, Tome 129 (2009) no. 9, pp. 2021-2034 | Zbl 1221.11188

[7] Tatsushi Tanaka; Noriko Wakabayashi An algebraic proof of the cyclic sum formula for multiple zeta values, J. Algebra, Tome 323 (2010) no. 3, pp. 766-778 | Zbl 1231.11106

[8] Tatsushi Tanaka; Noriko Wakabayashi Kawashima’s relations for interpolated multiple zeta values, J. Algebra, Tome 447 (2016), pp. 424-431 | Zbl 1370.11104

[9] Shuji Yamamoto Multiple zeta-star values and multiple integrals, RIMS Kôkyûroku Bessatsu, Tome B68 (2017), pp. 3-14 | Zbl 06937986