Identities on Bell polynomials and Sheffer sequences. (English) Zbl 1221.05018
The paper gives two new characterizations of Sheffer sequences in terms of exponential partial Bell polynomials. This yields connection between Sheffer sequences and Riordan arrays. Several general identities are established for Bell polynomials and Sheffer sequences, which specialize to elegant identities for associated sequences and cross sequences.
Reviewer: László A. Székely (Columbia)
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 11B99 | Sequences and sets |
Keywords:
associated sequences; Bell polynomials; cross sequences; Riordan arrays; Sheffer sequences; combinatorial identitiesReferences:
| [1] | Abbas, M.; Bouroubi, S., On new identities for Bell’s polynomials, Discrete Math., 293, 1-3, 5-10 (2005) · Zbl 1063.05014 |
| [2] | Charalambides, C. A., (Enumerative Combinatorics. Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications (2002), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL) · Zbl 1001.05001 |
| [3] | Comtet, L., Advanced Combinatorics (1974), D. Reidel Publishing Co.: D. Reidel Publishing Co. Dordrecht · Zbl 0283.05001 |
| [4] | Egorychev, G. P.; Zima, E. V., Decomposition and group theoretic characterization of pairs of inverse relations of the Riordan type, Acta Appl. Math., 85, 1-3, 93-109 (2005) · Zbl 1074.05012 |
| [5] | He, T.-X.; Hsu, L. C.; Shiue, P. J.-S., The Sheffer group and the Riordan group, Discrete Appl. Math., 155, 15, 1895-1909 (2007) · Zbl 1123.05007 |
| [6] | Merlini, D.; Sprugnoli, R.; Verri, M. C., Lagrange inversion: when and how, Acta Appl. Math., 94, 3, 233-249 (2006) · Zbl 1108.05008 |
| [7] | Merlini, D.; Sprugnoli, R.; Verri, M. C., The method of coefficients, Amer. Math. Monthly, 114, 1, 40-57 (2007) · Zbl 1191.05006 |
| [8] | Niven, I., Formal power series, Amer. Math. Monthly, 76, 871-889 (1969) · Zbl 0184.29603 |
| [9] | J. Riordan, Combinatorial Identities, Reprint of the 1968 original, Robert E. Krieger Publishing Co., Huntington, N Y, 1979; J. Riordan, Combinatorial Identities, Reprint of the 1968 original, Robert E. Krieger Publishing Co., Huntington, N Y, 1979 |
| [10] | Roman, S., The Umbral Calculus (1984), Academic Press, Inc.: Academic Press, Inc. New York · Zbl 0536.33001 |
| [11] | Roman, S.; Rota, G.-C., The umbral calculus, Adv. Math., 27, 2, 95-188 (1978) · Zbl 0375.05007 |
| [12] | Rota, G.-C.; Kahaner, D.; Odlyzko, A., On the foundations of combinatorial theory. VIII. Finite operator calculus, J. Math. Anal. Appl., 42, 684-760 (1973) · Zbl 0267.05004 |
| [13] | Shapiro, L. W.; Getu, S.; Woan, W. J.; Woodson, L. C., The Riordan group, Discrete Appl. Math., 34, 1-3, 229-239 (1991) · Zbl 0754.05010 |
| [14] | Sprugnoli, R., Riordan arrays and combinatorial sums, Discrete Math., 132, 1-3, 267-290 (1994) · Zbl 0814.05003 |
| [15] | Sprugnoli, R., Riordan arrays and the Abel-Gould identity, Discrete Math., 142, 1-3, 213-233 (1995) · Zbl 0832.05007 |
| [16] | Wang, W.; Wang, T., Generalized Riordan arrays, Discrete Math. (2008) · Zbl 1158.05008 |
| [17] | Yang, S.-l., Some identities involving the binomial sequences, Discrete Math., 308, 1, 51-58 (2008) · Zbl 1127.05006 |
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