The noncentral version of the Whitney numbers: a comprehensive study. (English) Zbl 1425.11047
Summary: This paper is a comprehensive study of a certain generalization of Whitney-type and Stirling-type numbers which unifies the classical Whitney numbers, the translated Whitney numbers, the classical Stirling numbers, and the noncentral Stirling (or \(r\)-Stirling) numbers. Several identities, applications, and occurrences are also presented.
Online Encyclopedia of Integer Sequences:
Least positive sequence with Hankel transform {1,1,1,1,1,...}.References:
| [1] | Dowling, T. A., A class of geometric lattices based on finite groups, Journal of Combinatorial Theory. Series B, 15, 61-86 (1973) · Zbl 0247.05019 · doi:10.1016/s0095-8956(73)80007-3 |
| [2] | Benoumhani, M., On Whitney numbers of Dowling lattices, Discrete Mathematics, 159, 1-3, 13-33 (1996) · Zbl 0861.05004 · doi:10.1016/0012-365x(95)00095-e |
| [3] | Benoumhani, M., On some numbers related to Whitney numbers of Dowling lattices, Advances in Applied Mathematics, 19, 1, 106-116 (1997) · Zbl 0876.05001 · doi:10.1006/aama.1997.0529 |
| [4] | Benoumhani, M., Log-concavity of Whitney numbers of Dowling lattices, Advances in Applied Mathematics, 22, 2, 186-189 (1999) · Zbl 0918.05003 · doi:10.1006/aama.1998.0621 |
| [5] | Belbachir, H.; Bousbaa, I. E., Translated Whitney and \(r\)-Whitney numbers: a combinatorial approach, Journal of Integer Sequences, 16, article 13.8.6 (2013) · Zbl 1292.05050 |
| [6] | Mangontarum, M. M.; Macodi-Ringia, A. P.; Abdulcarim, N. S., The translated Dowling polynomials and numbers, International Scholarly Research Notices, 2014 (2014) · Zbl 1460.05014 · doi:10.1155/2014/678408 |
| [7] | Mangontarum, M. M.; Dibagulun, A., On the translated Whitney numbers and their combinatorial properties, British Journal of Applied Science and Technology, 11, 5, 1-15 (2015) · doi:10.9734/bjast/2015/19973 |
| [8] | Callan, D., Cesàro’s Integral Formula for the Bell Numbers (Corrected) (2005) |
| [9] | Mező, I., A new formula for the Bernoulli polynomials, Results in Mathematics, 58, 3-4, 329-335 (2010) · Zbl 1237.11010 · doi:10.1007/s00025-010-0039-z |
| [10] | Cheon, G.-S.; Jung, J.-H., r-Whitney numbers of Dowling lattices, Discrete Mathematics, 312, 15, 2337-2348 (2012) · Zbl 1246.05009 · doi:10.1016/j.disc.2012.04.001 |
| [11] | Mező, I.; Ramirez, J. L., The linear algebra of the \(r\)-Whitney matices, Integral Transforms and Special Functions, 26, 3, 213-225 (2015) · Zbl 1371.11062 · doi:10.1080/10652469.2014.984180 |
| [12] | Koutras, M., Non-central Stirling numbers and some applications, Discrete Mathematics, 42, 1, 73-89 (1982) · Zbl 0506.10009 · doi:10.1016/0012-365x(82)90056-5 |
| [13] | Broder, A. Z., The \(r\)-Stirling numbers, Discrete Mathematics, 49, 3, 241-259 (1984) · Zbl 0535.05006 · doi:10.1016/0012-365x(84)90161-4 |
| [14] | Graham, R. L.; Knuth, D. E.; Patashnik, O., Concrete Mathematics (1989), Reading, Mass, USA: Addison-Wesley, Reading, Mass, USA · Zbl 0668.00003 |
| [15] | Pan, J., Matrix decomposition of the unified generalized Stirling numbers and inversion of the generalized factorial matrices, Journal of Integer Sequences, 15, article 12.6.6 (2012) · Zbl 1291.11055 |
| [16] | Hsu, L. C.; Shiue, P. J., A unified approach to generalized Stirling numbers, Advances in Applied Mathematics, 20, 3, 366-384 (1998) · Zbl 0913.05006 · doi:10.1006/aama.1998.0586 |
| [17] | Rahmani, M., Some results on Whitney numbers of Dowling lattices, Arab Journal of Mathematical Sciences, 20, 1, 11-27 (2014) · Zbl 1377.11032 · doi:10.1016/j.ajmsc.2013.02.002 |
| [18] | Tanny, S., On some numbers related to the Bell numbers, Canadian Mathematical Bulletin, 17, 733-738 (1975) · Zbl 0304.10007 · doi:10.4153/cmb-1974-132-8 |
| [19] | Rota, G.-C., The number of partitions of a set, The American Mathematical Monthly, 71, 498-504 (1964) · Zbl 0121.01803 · doi:10.2307/2312585 |
| [20] | Chen, K.-W., Identities from the binomial transform, Journal of Number Theory, 124, 1, 142-150 (2007) · Zbl 1120.05007 · doi:10.1016/j.jnt.2006.07.015 |
| [21] | Sloane, N. J., Least positive sequence with Hankel transform \(\left``\{1,1, 1,1, 1, \ldots\right``\}\) |
| [22] | Layman, J. W., The Hankel transform and some of its properties, Journal of Integer Sequences, 4, 1 (2001) · Zbl 0978.15022 |
| [23] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1972), New York, NY, USA: Dover, New York, NY, USA · Zbl 0543.33001 |
| [24] | Mező, I., The \(r\)-Bell numbers, Journal of Integer Sequences, 14, 1 (2011) · Zbl 1205.05017 |
| [25] | Corcino, R. B.; Corcino, C. B., On generalized Bell polynomials, Discrete Dynamics in Nature and Society, 2011 (2011) · Zbl 1272.11039 · doi:10.1155/2011/623456 |
| [26] | Comtet, L., Advanced Combinatorics (1974), Dordrecht, Netherlands: D. Reidel, Dordrecht, Netherlands · Zbl 0283.05001 |
| [27] | Cesàro, M., Sur une équation aux différences melées, Nouvelles Annales de Mathématiques, 4, 36-40 (1883) |
| [28] | Stirling, J., Methodus Differentialissme Tractus de Summatione et Interpolatione Serierum Infinitarum (1730), London, UK |
| [29] | Privault, N., Generalized Bell polynomials and the combinatorics of Poisson central moments, Electronic Journal of Combinatorics, 18, 1, 1-10 (2011) · Zbl 1215.11017 |
| [30] | Mangontarum, M. M.; Corcino, R. B., The generalized factorial moments in terms of a Poisson random variable, GSTF Journal of Mathematics, Statistics and Operations Research, 2, 1, 64-67 (2013) |
| [31] | Mező, I., On the maximum of \(r\)-Stirling numbers, Advances in Applied Mathematics, 41, 3, 293-306 (2008) · Zbl 1165.11023 · doi:10.1016/j.aam.2007.11.002 |
| [32] | Corcino, R. B.; Corcino, C. B., On the maximum of generalized Stirling numbers, Utilitas Mathematica, 86, 241-256 (2011) · Zbl 1294.05013 |
| [33] | Corcino, R. B.; Corcino, C. B.; Aranas, P. B., The peak of noncentral Stirling numbers of the first kind, International Journal of Mathematics and Mathematical Sciences, 2015 (2015) · Zbl 1393.11022 · doi:10.1155/2015/982812 |
| [34] | Corcino, C. B.; Corcino, R. B., Asymptotic estimates for second kind generalized stirling numbers, Journal of Applied Mathematics, 2013 (2013) · Zbl 1283.11045 · doi:10.1155/2013/918513 |
| [35] | Corcino, C. B.; Corcino, R. B., An asymptotic formula for r-Bell numbers with real arguments, ISRN Discrete Mathematics, 2013 (2013) · Zbl 1283.11045 · doi:10.1155/2013/274697 |
| [36] | Corcino, C. B.; Corcino, R. B.; Acala, N., Asymptotic estimates for r-Whitney numbers of the second kind, Journal of Applied Mathematics, 2014 (2014) · Zbl 1406.11020 · doi:10.1155/2014/354053 |
| [37] | Corcino, C. B.; Corcino, R. B.; Gasparin, R. J., Equivalent asymptotic formulas of second kind r-Whitney numbers, Integral Transforms and Special Functions, 26, 3, 192-202 (2015) · Zbl 1371.11061 · doi:10.1080/10652469.2014.979410 |
| [38] | Corcino, C. B.; Corcino, R. B.; Ontolan, J. M.; Perez-Fernandez, C. M.; Cantallopez, E. R., The Hankel transform of \(q\)-noncentral Bell numbers, International Journal of Mathematics and Mathematical Sciences, 2015 (2015) · Zbl 1393.11021 · doi:10.1155/2015/417327 |
| [39] | de Médicis, A.; Leroux, P., Generalized Stirling numbers, convolution formulae and p; \(q\)-analogues, Canadian Journal of Mathematics, 47, 3, 474-499 (1995) · Zbl 0831.05005 · doi:10.4153/cjm-1995-027-x |
| [40] | El-Desouky, B. S., The multiparameter noncentral Stirling numbers, The Fibonacci Quarterly, 32, 218-225 (1994) · Zbl 0802.05003 |
| [41] | Corcino, R. B.; Mangontarum, M. M., On multiparameter \(q\)-noncentral Stirling and Bell numbers, Ars Combinatoria, 118, 201-220 (2015) · Zbl 1349.11047 |
| [42] | Mangontarum, M. M.; Katriel, J., On \(q\)-boson operators and \(q\)-analogues of the \(r\)-Whitney and \(r\)-dowling numbers, Journal of Integer Sequences, 18, article 15.9.8 (2015) · Zbl 1402.11031 |
| [43] | Arik, M.; Coon, D. D., Hilbert spaces of analytic functions and generalized coherent states, Journal of Mathematical Physics, 17, 4, 524-527 (1976) · Zbl 0941.81549 · doi:10.1063/1.522937 |
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