×

zbMATH — the first resource for mathematics

Recurrence relations for the Sheffer sequences. (English) Zbl 1251.05015
Summary: In this paper, using the production matrix of an exponential Riordan array \([g(t), f(t)]\), we give a recurrence relation for the Sheffer sequence for the ordered pair \((g(t), f(t))\). We also develop a new determinant representation for the general term of the Sheffer sequence. As applications, determinant expressions for some classical Sheffer polynomial sequences are derived.

MSC:
05A40 Umbral calculus
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Barry, P., On a family of generalized Pascal triangles defined by exponential Riordan arrays, J. integer seq., 10, (2007), Article 07.3.5 · Zbl 1158.05004
[2] Comtet, L., Advanced combinatorics, (1974), D. Reidel Publishing Co. Dordrecht
[3] Costabile, F.A.; Longo, E., A determinant approach to Appell polynomials, J. comput. appl. math., 234, 1528-1542, (2010) · Zbl 1200.33020
[4] Deutsch, E.; Ferrari, L.; Rinaldi, S., Production matrices, Adv. in appl. math., 34, 101-122, (2005) · Zbl 1064.05012
[5] Deutsch, E.; Ferrari, L.; Rinaldi, S., Production matrices and Riordan array, Ann. comb., 13, 65-85, (2009) · Zbl 1229.05015
[6] He, T.X.; Sprugnoli, R., Sequence characterization of Riordan arrays, Discrete math., 309, 3962-3974, (2009) · Zbl 1228.05014
[7] Luzón, A.; Morón, M., Recurrence relations for polynomial sequences via Riordan matrices, Linear algebra appl., 433, 1422-1446, (2010) · Zbl 1250.11029
[8] Merlini, D.; Rogers, D.G.; Sprugnoli, R.; Verri, M.C., On some alternative characterizations of Riordan arrays, Canad. J. math., 49, 2, 301-320, (1997) · Zbl 0886.05013
[9] Munarini, E.; Tprri, D., Cayley continuants, Theoret. comput. sci., 347, 353-369, (2005) · Zbl 1089.15010
[10] Peart, P.; Woan, W.-J., Generating functions via Hankel and Stieltjes matrices, J. integer seq., 3, 2, (2000), Article 00.2.1 · Zbl 0961.15018
[11] Roman, S., The umbral calculus, (1984), Academic Press Inc. · Zbl 0536.33001
[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., The Riordan group, Discrete appl. math., 34, 229-239, (1991) · Zbl 0754.05010
[14] Shapiro, L., Bijections and the Riordan group, Theoret. comput. sci., 307, 403-413, (2003) · Zbl 1048.05008
[15] Sprugnoli, R., Riordan arrays and the abel – gould identity, Discrete math., 142, 213-233, (1995) · Zbl 0832.05007
[16] Sprugnoli, R., Riordan arrays and combinatorial sums, Discrete math., 132, 267-290, (1994) · Zbl 0814.05003
[17] Wang, W.; Wang, T., Generalized Riordan arrays, Discrete math., 308, 6466-6500, (2008) · Zbl 1158.05008
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.