On a duality formula for certain sums of values of poly-Bernoulli polynomials and its application
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 1, pp. 203-218.

We prove a duality formula for certain sums of values of poly-Bernoulli polynomials which generalizes dualities for poly-Bernoulli numbers. We first compute two types of generating functions for these sums, from which the duality formula is apparent. Secondly we give an analytic proof of the duality from the viewpoint of our previous study of zeta functions of Arakawa–Kaneko type. As an application, we give a formula that relates poly-Bernoulli numbers to the Genocchi numbers.

Nous prouvons une formule de dualité pour certaines sommes de valeurs de polynômes poly-Bernoulli qui généralise les dualités pour les nombres de poly-Bernoulli. On calcule d’abord deux types de fonctions génératrices de ces sommes, dont la formule de dualité est apparente. Ensuite, nous donnons une preuve analytique de la dualité du point de vue de notre étude précédente de fonctions zêta de type Arakawa–Kaneko. Comme application, nous donnons une formule qui relie les nombres de poly-Bernoulli aux nombres de Genocchi.

Accepted:
Published online:
DOI: 10.5802/jtnb.1023
Classification: 11B68, 11M32
Keywords: Poly-Bernoulli numbers, Poly-Bernoulli polynomials, Arakawa–Kaneko zeta-functions, Genocchi numbers
Masanobu Kaneko 1; Fumi Sakurai 2; Hirofumi Tsumura 3

1 Faculty of Mathematics Kyushu University Motooka 744, Nishi-ku Fukuoka 819-0395, Japan
2 Graduate School of Mathematics Kyushu University Motooka 744, Nishi-ku Fukuoka 819-0395, Japan
3 Department of Mathematics and Information Sciences Tokyo Metropolitan University 1-1, Minami-Ohsawa, Hachioji Tokyo 192-0397, Japan
@article{JTNB_2018__30_1_203_0,
author = {Masanobu Kaneko and Fumi Sakurai and Hirofumi Tsumura},
title = {On a duality formula for certain sums of values of {poly-Bernoulli} polynomials and its application},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {203--218},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {30},
number = {1},
year = {2018},
doi = {10.5802/jtnb.1023},
zbl = {1435.11043},
mrnumber = {3809716},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1023/}
}
TY  - JOUR
AU  - Masanobu Kaneko
AU  - Fumi Sakurai
AU  - Hirofumi Tsumura
TI  - On a duality formula for certain sums of values of poly-Bernoulli polynomials and its application
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 203
EP  - 218
VL  - 30
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1023/
DO  - 10.5802/jtnb.1023
LA  - en
ID  - JTNB_2018__30_1_203_0
ER  - 
%0 Journal Article
%A Masanobu Kaneko
%A Fumi Sakurai
%A Hirofumi Tsumura
%T On a duality formula for certain sums of values of poly-Bernoulli polynomials and its application
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 203-218
%V 30
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1023/
%R 10.5802/jtnb.1023
%G en
%F JTNB_2018__30_1_203_0
Masanobu Kaneko; Fumi Sakurai; Hirofumi Tsumura. On a duality formula for certain sums of values of poly-Bernoulli polynomials and its application. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 1, pp. 203-218. doi : 10.5802/jtnb.1023. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1023/

[1] Tsuneo Arakawa; Tomoyoshi Ibukiyama; Masanobu Kaneko Bernoulli Numbers and Zeta Functions, Springer Monographs in Mathematics, Springer, 2014, xi+274 pages | Zbl

[2] Tsuneo Arakawa; Masanobu Kaneko Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J., Volume 153 (1999), pp. 189-209 | DOI | MR | Zbl

[3] Tsuneo Arakawa; Masanobu Kaneko On poly-Bernoulli numbers, Comment. Math. Univ. St. Pauli, Volume 48 (1999) no. 2, pp. 159-167 | MR | Zbl

[4] Beáta Bényi; Péter Hajnal Combinatorics of poly-Bernoulli numbers, Stud. Sci. Math. Hung., Volume 52 (2015) no. 4, pp. 537-558 | MR | Zbl

[5] Chad Brewbaker A combinatorial interpretation of the Poly-Bernoulli numbers and two Fermat analogues, Integers, Volume 8 (2008) no. 1 (article A02) | MR | Zbl

[6] Peter J. Cameron; C. A. Glass; Raphael Schumacher Acyclic orientations and poly-Bernoulli numbers (2014) (https://arxiv.org/abs/1412.3685)

[7] Marc-Antoine Coppo; Bernard Candelpergher The Arakawa-Kaneko zeta function, Ramanujan J., Volume 22 (2010) no. 2, pp. 153-162 | DOI | MR | Zbl

[8] Yoshinori Hamahata; H. Masubuchi Recurrence formulae for multi-poly-Bernoulli numbers, Integers, Volume 7 (2007) no. 1, article A46 pages | MR | Zbl

[9] Yoshinori Hamahata; H. Masubuchi Special multi-poly-Bernoulli numbers, J. Integer Seq., Volume 10 (2007) no. 4 (article 07.4.1) | MR | Zbl

[10] Masanobu Kaneko Poly-Bernoulli numbers, J. Théor. Nombres Bordx., Volume 9 (1997) no. 1, pp. 221-228 | DOI | Numdam | MR | Zbl

[11] Masanobu Kaneko Poly-Bernoulli numbers and related zeta functions, Algebraic and analytic aspects of zeta functions and $L$-functions (MSJ Memoirs), Volume 21, Mathematical Society of Japan, 2010, pp. 73-85 | DOI | MR | Zbl

[12] Masanobu Kaneko; Hirofumi Tsumura Multi-poly-Bernoulli numbers and related zeta functions, Nagoya Math. J. (2017) (https://doi.org/10.1017/nmj.2017.16) | DOI | Zbl

[13] François Édouard Anatole Lucas Théorie des nombres. Vol I Le calcul des nombres entiers. Le calcul des nombres rationnels. La divisibilité arithmétique, Gauthier-Villars et Fils, 1891, xxxiv+520 pages | Zbl

[14] Richard P. Stanley Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999, xii+581 pages | MR | Zbl

[15] E. Takeda On Multi-Poly-Bernoulli numbers Master’s Thesis, Kyushu University (Japan), 2013

[16] Lawrence C. Washington Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, 83, Springer, 1997, xiv+487 pages | MR | Zbl

Cited by Sources: