In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
Dans cet article, nous introduisons des modèles combinatoires pour les analogues en plusieurs variables des polynômes de Bernoulli et des nombres d’Euler de deux types. Comme application, nous donnons des preuves combinatoires de certaines identités faisant intervenir les polynômes de Bernoulli généralisés.
Revised:
Accepted:
Published online:
Keywords: Poly-Bernoulli polynomial, poly-Euler number
@article{JTNB_2022__34_3_917_0, author = {B\'enyi, Be\'ata and Matsusaka, Toshiki}, title = {Combinatorial aspects of {poly-Bernoulli} polynomials and {poly-Euler} numbers}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {917--939}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {34}, number = {3}, year = {2022}, doi = {10.5802/jtnb.1234}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jtnb.1234/} }
TY - JOUR AU - Bényi, Beáta AU - Matsusaka, Toshiki TI - Combinatorial aspects of poly-Bernoulli polynomials and poly-Euler numbers JO - Journal de théorie des nombres de Bordeaux PY - 2022 SP - 917 EP - 939 VL - 34 IS - 3 PB - Société Arithmétique de Bordeaux UR - http://www.numdam.org/articles/10.5802/jtnb.1234/ DO - 10.5802/jtnb.1234 LA - en ID - JTNB_2022__34_3_917_0 ER -
%0 Journal Article %A Bényi, Beáta %A Matsusaka, Toshiki %T Combinatorial aspects of poly-Bernoulli polynomials and poly-Euler numbers %J Journal de théorie des nombres de Bordeaux %D 2022 %P 917-939 %V 34 %N 3 %I Société Arithmétique de Bordeaux %U http://www.numdam.org/articles/10.5802/jtnb.1234/ %R 10.5802/jtnb.1234 %G en %F JTNB_2022__34_3_917_0
Bényi, Beáta; Matsusaka, Toshiki. Combinatorial aspects of poly-Bernoulli polynomials and poly-Euler numbers. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 917-939. doi : 10.5802/jtnb.1234. http://www.numdam.org/articles/10.5802/jtnb.1234/
[1] Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J., Volume 153 (1999), pp. 189-209 | DOI | MR | Zbl
[2] Polylogarithms and poly-Bernoulli polynomials, Kyushu J. Math., Volume 65 (2011) no. 1, pp. 15-24 | DOI | MR | Zbl
[3] Combinatorial properties of poly-Bernoulli relatives, Integers, Volume 17 (2017), A31, 26 pages | MR | Zbl
[4] On the combinatorics of symmetrized poly-Bernoulli numbers, Electron. J. Comb., Volume 28 (2021) no. 1, 1.47, 20 pages | DOI | MR | Zbl
[5] A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, Integers, Volume 8 (2008), A02, 9 pages | MR | Zbl
[6] The -Stirling numbers, Discrete Math., Volume 49 (1984) no. 3, pp. 241-259 | DOI | MR | Zbl
[7] Weighted Stirling numbers of the first and second kind. I, Fibonacci Q., Volume 18 (1980) no. 2, pp. 147-162 | MR | Zbl
[8] Weighted Stirling numbers of the first and second kind. II, Fibonacci Q., Volume 18 (1980) no. 3, pp. 242-257 | MR | Zbl
[9] The Arakawa-Kaneko zeta function, Ramanujan J., Volume 22 (2010) no. 2, pp. 153-162 | DOI | MR | Zbl
[10] The algebraic complexity of maximum likelihood estimation for bivariate missing data, Algebraic and geometric methods in statistics, Cambridge University Press, 2010, pp. 123-133 | MR
[11] Poly-Bernoulli numbers, J. Théor. Nombres Bordeaux, Volume 9 (1997) no. 1, pp. 221-228 | DOI | Numdam | MR | Zbl
[12] Generalized harmonic numbers via poly-Bernoulli polynomials (2021) (https://arxiv.org/abs/2008.00284v2)
[13] Johann Faulhaber and sums of powers, Math. Comput., Volume 61 (1993) no. 203, pp. 277-294 | DOI | MR | Zbl
[14] Complementary Euler numbers, Period. Math. Hung., Volume 75 (2017) no. 2, pp. 302-314 | DOI | MR | Zbl
[15] On poly-Euler numbers of the second kind, Algebraic number theory and related topics 2016 (RIMS Kôkyûroku Bessatsu), Volume B77, Research Institute for Mathematical Sciences, 2020, pp. 143-158 | Zbl
[16] Some relationships between poly-Cauchy numbers and poly-Bernoulli numbers, Ann. Math. Inform., Volume 41 (2013), pp. 99-105 | MR | Zbl
[17] Hypergeometric Euler numbers, AIMS Math., Volume 5 (2020) no. 2, pp. 1284-1303 | DOI | MR | Zbl
[18] Turán type inequalities for the partial sums of the generating functions of Bernoulli and Euler numbers, Math. Nachr., Volume 285 (2012) no. 17-18, pp. 2129-2156 | DOI | MR | Zbl
[19] On the hyper-sums of powers of integers, Miskolc Math. Notes, Volume 18 (2017) no. 1, pp. 307-314 | DOI | MR | Zbl
[20] Rigidly Foldable Quadrilateral Meshes From Angle Arrays, J. Mech. Robot., Volume 10 (2018) no. 2, 021004, 11 pages | DOI
[21] Noncommutative Biology: Sequential Regulation of Complex Networks, PLoS Comput. Biol., Volume 12 (2016) no. 8, pp. 1-36
[22] On the parity of poly-Euler numbers, Algebraic number theory and related topics 2010 (RIMS Kôkyûroku Bessatsu), Volume B32, Research Institute for Mathematical Sciences, 2012, pp. 271-278 | MR | Zbl
[23] On poly-Euler numbers, J. Aust. Math. Soc., Volume 103 (2017) no. 1, pp. 126-144 | DOI | MR | Zbl
[24] Combinatorial properties of matrices of zeros and ones, Can. J. Math., Volume 9 (1957), pp. 371-377 | DOI | MR | Zbl
[25] On generalized poly-Bernoulli numbers and related -functions, J. Number Theory, Volume 132 (2012) no. 1, pp. 156-170 | DOI | MR | Zbl
[26] The on-line encyclopedia of integer sequences (Available at https://oeis.org)
[27] Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, 1997, xii+325 pages (with a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original) | DOI | MR | Zbl
Cited by Sources: