×

On a duality formula for certain sums of values of poly-Bernoulli polynomials and its application. (English. French summary) Zbl 1435.11043

Summary: 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.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11M32 Multiple Dirichlet series and zeta functions and multizeta values
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Tsuneo Arakawa, Tomoyoshi Ibukiyama & Masanobu Kaneko, Bernoulli Numbers and Zeta Functions, Springer Monographs in Mathematics, Springer, 2014 · Zbl 1312.11015
[2] Tsuneo Arakawa & Masanobu Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J.153 (1999), p. 189-209 · Zbl 0932.11055 · doi:10.1017/S0027763000006954
[3] Tsuneo Arakawa & Masanobu Kaneko, On poly-Bernoulli numbers, Comment. Math. Univ. St. Pauli48 (1999), p. 159-167 · Zbl 0994.11009
[4] Beáta Bényi & Péter Hajnal, Combinatorics of poly-Bernoulli numbers, Stud. Sci. Math. Hung.52 (2015), p. 537-558 · Zbl 1374.05002
[5] Chad Brewbaker, A combinatorial interpretation of the Poly-Bernoulli numbers and two Fermat analogues, Integers8 (2008) · Zbl 1165.11022
[6] Peter J. Cameron, C. A. Glass & Raphael Schumacher, “Acyclic orientations and poly-Bernoulli numbers”, , 2014
[7] Marc-Antoine Coppo & Bernard Candelpergher, The Arakawa-Kaneko zeta function, Ramanujan J.22 (2010), p. 153-162 · Zbl 1230.11106 · doi:10.1007/s11139-009-9205-x
[8] Yoshinori Hamahata & H. Masubuchi, Recurrence formulae for multi-poly-Bernoulli numbers, Integers7 (2007) · Zbl 1148.11010
[9] Yoshinori Hamahata & H. Masubuchi, Special multi-poly-Bernoulli numbers, J. Integer Seq.10 (2007) · Zbl 1140.11310
[10] Masanobu Kaneko, Poly-Bernoulli numbers, J. Théor. Nombres Bordx.9 (1997), p. 221-228 · Zbl 0887.11011 · doi:10.5802/jtnb.197
[11] Masanobu Kaneko, Poly-Bernoulli numbers and related zeta functions, Algebraic and analytic aspects of zeta functions and \(L\)-functions, MSJ Memoirs 21, Mathematical Society of Japan, 2010, p. 73-85 · Zbl 1269.11080
[12] Masanobu Kaneko & Hirofumi Tsumura, Multi-poly-Bernoulli numbers and related zeta functions, Nagoya Math. J. (2017) · Zbl 1453.11035 · doi:10.1017/nmj.2017.16
[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 · Zbl 0464.10001
[14] Richard P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, 1999
[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 - ISSN : 2118-8572 (online) 1246-7405 (print) - Société Arithmétique de Bordeaux · Zbl 0966.11047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.