On finite Carlitz multiple polylogarithms

Chieh-Yu Chang; Yoshinori Mishiba

doi:10.5802/jtnb.1011

Chieh-Yu Chang ¹; Yoshinori Mishiba ²

¹ Department of Mathematics National Tsing Hua University Hsinchu City 30042, Taiwan R.O.C.
² Department of Life, Environment and Materials Science Fukuoka Institute of Technology, Japan

Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 1049-1058.

Abstract
Résumé

In this paper, we define finite Carlitz multiple polylogarithms and show that every finite multiple zeta value over the rational function field $𝔽_{q} (θ)$ is an $𝔽_{q} (θ)$ -linear combination of finite Carlitz multiple polylogarithms at integral points. It is completely compatible with the formula for Thakur MZV’s established in [6].

Dans cet article nous définissons la notion de polylogarithme multiple fini de Carlitz et montrons que chaque valeur de zêta multiple finie définie sûr un corps de fonctions rationnelles $𝔽_{q} (θ)$ est une combinaison linéaire des valeurs des polylogarithmes multiples finis de Carlitz evalués en des points entiers. Cela est complètement compatible avec la formule des MZVs de Thakur établie dans [6].

Received: 2016-11-07
Revised: 2017-05-08
Accepted: 2017-05-11
Published online: 2017-12-12

DOI: 10.5802/jtnb.1011

Classification: 11R58, 11M38
Keywords: Finite Carlitz multiple polylogarithms, finite multiple zeta values, Anderson–Thakur polynomials

Author's affiliations:

Chieh-Yu Chang ¹; Yoshinori Mishiba ²

¹ Department of Mathematics National Tsing Hua University Hsinchu City 30042, Taiwan R.O.C.
² Department of Life, Environment and Materials Science Fukuoka Institute of Technology, Japan

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Chieh-Yu Chang and Yoshinori Mishiba},
     title = {On finite {Carlitz} multiple polylogarithms},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {1049--1058},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {3},
     year = {2017},
     doi = {10.5802/jtnb.1011},
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Chieh-Yu Chang; Yoshinori Mishiba. On finite Carlitz multiple polylogarithms. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 1049-1058. doi : 10.5802/jtnb.1011. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1011/

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