The partial \(r\)-Bell polynomials. (English) Zbl 1387.05018
Summary: In this paper, we show that the \(r\)-Stirling numbers of both kinds, the \(r\)-Whitney numbers of both kinds, the \(r\)-Lah numbers and the \(r\)-Whitney-Lah numbers form particular cases of a family of polynomials forming a generalization of the partial Bell polynomials. We deduce the generating functions of several restrictions of these numbers. In addition, a new combinatorial interpretations is presented for the \(r\)-Whitney numbers and the \(r\)-Whitney-Lah numbers.
Keywords:
partial Bell polynomials; \(r\)-Bell polynomials; recurrence relations; Stirling numbers; Whitney numbers; probabilistic interpretationReferences:
| [1] | Bell, E.T.: Exponential polynomials. Ann. Math. 35, 258-277 (1934) · Zbl 0009.21202 · doi:10.2307/1968431 |
| [2] | Broder, A.Z.: The \[r\] r-Stirling numbers. Discrete Math. 49, 241-259 (1984) · Zbl 0535.05006 · doi:10.1016/0012-365X(84)90161-4 |
| [3] | Chrysaphinou, O.: On Touchard polynomials. Discrete Math. 54(2), 143-152 (1985) · Zbl 0567.33013 · doi:10.1016/0012-365X(85)90075-5 |
| [4] | Cheon, G.S., Jung, J.H.: r-Whitney numbers of Dowling lattices. Discrete Math. 308, 2450-2459 (2012) · Zbl 1246.05009 |
| [5] | Comtet, L.: Advanced Combinatorics. D. Reidel Publishing Company, Dordrecht (1974) · Zbl 0283.05001 · doi:10.1007/978-94-010-2196-8 |
| [6] | Corcino, C.B., Hsu, L.C., Tan, E.L.: Asymptotic approximations of \[r\] r-Stirling numbers. Approx. Theory Appl. (N.S.) 15, 13-25 (1999) · Zbl 0952.41019 |
| [7] | Maamra, M.S., Mihoubi, M.: Note on some restricted Stirling numbers of the second kind. C. R. Math. Acad. Sci. 354, 231-234 (2016) · Zbl 1387.05120 · doi:10.1016/j.crma.2015.12.003 |
| [8] | Mező, I.: A new formula for the Bernoulli polynomials. Results. Math. 58, 329-335 (2010) · Zbl 1237.11010 · doi:10.1007/s00025-010-0039-z |
| [9] | Mező, I.: On the maximum of \[r\] r-Stirling numbers. Adv. Appl. Math. 41, 293-306 (2008) · Zbl 1165.11023 · doi:10.1016/j.aam.2007.11.002 |
| [10] | Mező, I.: New properties of \[r\] r-Stirling series. Acta Math. Hungar. 119, 341-358 (2008) · Zbl 1174.11026 · doi:10.1007/s10474-007-7047-9 |
| [11] | Mező, I.: The \[r\] r-Bell numbers. J. Integer Seq. 14, Article 11.1.1, p 14 (2011) · Zbl 1205.05017 |
| [12] | Mihoubi, M.: Bell polynomials and binomial type sequences. Discrete Math. 308, 2450-2459 (2008) · Zbl 1147.05006 · doi:10.1016/j.disc.2007.05.010 |
| [13] | Mihoubi, M.: Partial Bell Polynomials and Inverse Relations. J. Integer Seq., 13, Article 10.4.5. (2010) · Zbl 1197.05005 |
| [14] | Mihoubi, M.: Polynômes multivariés de Bell et polynômes de type binomial. Thesis (Ph. D.) Université des sciences et de la technologie Houari-Boumediene, Alger (2008) · Zbl 0009.21202 |
| [15] | Mihoubi, M., Maamra, M.S.: The \[(r_1,\dots, r_p)\](r1,⋯,rp)-Stirling numbers of the second kind. Integers 12(5), 1047-1059 (2012) · Zbl 1310.11031 · doi:10.1515/integers-2012-0022 |
| [16] | Mihoubi, M., Reggane, L.: The \[s\] s-degenerate \[r\] r-Lah numbers. Ars Combin. 120, 333-340 (2015) · Zbl 1349.11018 |
| [17] | Port, D.: Polynomial maps with applications to combinatorics and probability theory. Thesis (Ph. D.) Massachusetts Institute of Technology, Dept. of Mathematics (1994) |
| [18] | Pitman, J.: Combinatorial stochastic processes. Lecture Notes in Mathematics, vol. 1875. Springer-Verlag, Berlin (2006) · Zbl 1103.60004 |
| [19] | Rahmani, M.: Some results on Whitney numbers of Dowling lattices. Arab J. Math. Sci. 20(1), 11-27 (2014) · Zbl 1377.11032 |
| [20] | Rahmani, M.: Generalized Stirling transform. Miskolc. Math. Notes 15(2), 677-690 (2014) · Zbl 1324.11024 |
| [21] | Wang, W., Wang, T.: General identities on Bell polynomials. Comput. Math. Appl. 58(1), 104-118 (2009) · Zbl 1189.33021 · doi:10.1016/j.camwa.2009.03.093 |
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.
