On a duality formula for certain sums of values of poly-Bernoulli polynomials and its application

Kaneko, Masanobu; Sakurai, Fumi; Tsumura, Hirofumi

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 1, pp. 203-218.

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.

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.

Classification : 11B68, 11M32
Mots clés : Poly-Bernoulli numbers, Poly-Bernoulli polynomials, Arakawa–Kaneko zeta-functions, Genocchi numbers

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