Combinatorial aspects of poly-Bernoulli polynomials and poly-Euler numbers. (English. French summary) Zbl 1509.05027
Summary: 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.
MSC:
| 05A19 | Combinatorial identities, bijective combinatorics |
| 11B68 | Bernoulli and Euler numbers and polynomials |
Software:
OEISReferences:
| [1] | Arakawa, Tsuneo; Kaneko, Masanobou, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J., 153, 189-209 (1999) · Zbl 0932.11055 · doi:10.1017/S0027763000006954 |
| [2] | Bayad, Abdelmejid; Hamahata, Yoshinori, Polylogarithms and poly-Bernoulli polynomials, Kyushu J. Math., 65, 1, 15-24 (2011) · Zbl 1268.11030 · doi:10.2206/kyushujm.65.15 |
| [3] | Bényi, Beáta; Hajnal, Péter, Combinatorial properties of poly-Bernoulli relatives, Integers, 17, 26 p. pp. (2017) · Zbl 1412.11034 |
| [4] | Bényi, Beáta; Matsusaka, Toshiki, On the combinatorics of symmetrized poly-Bernoulli numbers, Electron. J. Comb., 28, 1, 20 p. pp. (2021) · Zbl 1509.05026 · doi:10.37236/9743 |
| [5] | Brewbaker, Chad, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, Integers, 8, 9 p. pp. (2008) · Zbl 1165.11022 |
| [6] | Broder, Andrei Z., The \(r\)-Stirling numbers, Discrete Math., 49, 3, 241-259 (1984) · Zbl 0535.05006 · doi:10.1016/0012-365X(84)90161-4 |
| [7] | Carlitz, Leonard, Weighted Stirling numbers of the first and second kind. I, Fibonacci Q., 18, 2, 147-162 (1980) · Zbl 0428.05003 |
| [8] | Carlitz, Leonard, Weighted Stirling numbers of the first and second kind. II, Fibonacci Q., 18, 3, 242-257 (1980) · Zbl 0441.05003 |
| [9] | Coppo, Marc-Antoine; Candelpergher, Bernard, The Arakawa-Kaneko zeta function, Ramanujan J., 22, 2, 153-162 (2010) · Zbl 1230.11106 · doi:10.1007/s11139-009-9205-x |
| [10] | Hoşten, Serkan; Sullivant, Seth, Algebraic and geometric methods in statistics, The algebraic complexity of maximum likelihood estimation for bivariate missing data, 123-133 (2010), Cambridge University Press |
| [11] | Kaneko, Masanobou, Poly-Bernoulli numbers, J. Théor. Nombres Bordeaux, 9, 1, 221-228 (1997) · Zbl 0887.11011 |
| [12] | Kargın, Levent; Cenkci, Mehmet; Dil, Ayhan; Can, Mümün, Generalized harmonic numbers via poly-Bernoulli polynomials (2021) |
| [13] | Knuth, Donald E., Johann Faulhaber and sums of powers, Math. Comput., 61, 203, 277-294 (1993) · Zbl 0797.11026 · doi:10.2307/2152953 |
| [14] | Komatsu, Takao, Complementary Euler numbers, Period. Math. Hung., 75, 2, 302-314 (2017) · Zbl 1413.11126 · doi:10.1007/s10998-017-0199-7 |
| [15] | Komatsu, Takao, Algebraic number theory and related topics 2016, B77, On poly-Euler numbers of the second kind, 143-158 (2020), Research Institute for Mathematical Sciences · Zbl 1454.11048 |
| [16] | Komatsu, Takao; Luca, Florian, Some relationships between poly-Cauchy numbers and poly-Bernoulli numbers, Ann. Math. Inform., 41, 99-105 (2013) · Zbl 1289.11021 |
| [17] | Komatsu, Takao; Zhu, Huilin, Hypergeometric Euler numbers, AIMS Math., 5, 2, 1284-1303 (2020) · Zbl 1484.11077 · doi:10.3934/math.2020088 |
| [18] | Koumandos, Stamatis; Pedersen, Henrik Laurberg, Turán type inequalities for the partial sums of the generating functions of Bernoulli and Euler numbers, Math. Nachr., 285, 17-18, 2129-2156 (2012) · Zbl 1272.11038 · doi:10.1002/mana.201100299 |
| [19] | Laissaoui, Diffalah; Bounebirat, Fouad; Rahmani, Mourad, On the hyper-sums of powers of integers, Miskolc Math. Notes, 18, 1, 307-314 (2017) · Zbl 1399.11062 · doi:10.18514/MMN.2017.2030 |
| [20] | Lang, Robert J.; Howell, Larry, Rigidly Foldable Quadrilateral Meshes From Angle Arrays, J. Mech. Robot., 10, 2, 11 p. pp. (2018) · doi:10.1115/1.4038972 |
| [21] | Letsou, William; Cai, Long, Noncommutative Biology: Sequential Regulation of Complex Networks, PLoS Comput. Biol., 12, 8, 1-36 (2016) |
| [22] | Ohno, Yasuo; Sasaki, Yoshitaka, Algebraic number theory and related topics 2010, B32, On the parity of poly-Euler numbers, 271-278 (2012), Research Institute for Mathematical Sciences · Zbl 1334.11014 |
| [23] | Ohno, Yasuo; Sasaki, Yoshitaka, On poly-Euler numbers, J. Aust. Math. Soc., 103, 1, 126-144 (2017) · Zbl 1403.11016 · doi:10.1017/S1446788716000495 |
| [24] | Ryser, Herbert J., Combinatorial properties of matrices of zeros and ones, Can. J. Math., 9, 371-377 (1957) · Zbl 0079.01102 · doi:10.4153/CJM-1957-044-3 |
| [25] | Sasaki, Yoshitaka, On generalized poly-Bernoulli numbers and related \({L}\)-functions, J. Number Theory, 132, 1, 156-170 (2012) · Zbl 1268.11135 · doi:10.1016/j.jnt.2011.07.007 |
| [26] | Sloane, Neil J. A., The on-line encyclopedia of integer sequences · Zbl 1274.11001 |
| [27] | Stanley, Richard P., Enumerative combinatorics. Vol. 1, 49, xii+325 p. pp. (1997), Cambridge University Press · Zbl 0889.05001 · doi:10.1017/CBO9780511805967 |
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.
