On a duality formula for certain sums of values of poly-Bernoulli polynomials and its application
Masanobu Kaneko1; Fumi Sakurai2; Hirofumi Tsumura3
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
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.

Received:
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
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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/

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