Luzón, Ana; Morón, Manuel A. Recurrence relations for polynomial sequences via Riordan matrices. (English) Zbl 1250.11029 Linear Algebra Appl. 433, No. 7, 1422-1446 (2010). Summary: We give recurrence relations for any family of generalized Appell polynomials unifying so some known recurrences for many classical sequences of polynomials. Our main tool to get our goal is the Riordan group. We use the product of Riordan matrices to interpret some relationships between different polynomial families. Moreover using the Hadamard product of series we get a general recurrence relation for the polynomial sequences associated to the so called generalized umbral calculus. Cited in 21 Documents MSC: 11B83 Special sequences and polynomials 33C65 Appell, Horn and Lauricella functions 68W30 Symbolic computation and algebraic computation 68R05 Combinatorics in computer science 05A40 Umbral calculus 05E15 Combinatorial aspects of groups and algebras (MSC2010) Keywords:recurrence relation; Riordan matrix; generalized Appell polynomial; polynomial sequence of Riordan type; umbral calculus PDF BibTeX XML Cite \textit{A. Luzón} and \textit{M. A. Morón}, Linear Algebra Appl. 433, No. 7, 1422--1446 (2010; Zbl 1250.11029) Full Text: DOI arXiv OpenURL References: [1] Bell, E.T., The history of blissard’s symbolic method, with a sketch of its inventor’s life, Amer. math. monthly, 45, 7, 414-421, (1938) · Zbl 0019.38902 [2] Boas, R.P.; Buck, R.C., Polynomial expansions of analytic functions, (1964), Springer-Verlag · Zbl 0116.28105 [3] Boubaker, K.; Chaouachi, A.; Amlouk, M.; Bouzouita, H., Enhancement of pyrolysis spray disposal perfomance using thermal time-response to precursor uniform deposition, Eur. phys. J. appl. phys., 37, 105-109, (2007) [4] He, T.-X.; Hsu, L.C.; Shiue, P.J.-S., The Sheffer group and the Riordan group, Discrete appl. math., 155, 1895-1909, (2007) · Zbl 1123.05007 [5] Huang, I.C., Inverse relations and Schauder bases, J. combin. theory ser. A, 97, 203-224, (2002) · Zbl 0998.05007 [6] Knuth, D.E., Convolution polynomials, Math. J., 2, 67-78, (1992) [7] Labiadh, H.; Dada, M.; Awojoyogbe, B.; Mahmoud, B.; Bannour, A., Establishment of an ordinary generating function and a christoffel – darboux type first-order differential equation for the heat equation related boubaker – turki polynomials, Differential equations control process., n°1, 52-66, (2008) · Zbl 07038878 [8] A. Luzón, Iterative processes related to Riordan arrays: the reciprocation and the inversion of power series, preprint. [9] Luzón, A.; Morón, M.A., Ultrametrics, banach’s fixed point theorem and the Riordan group, Discrete appl. math., 156, 2620-2635, (2008) · Zbl 1152.54032 [10] Luzón, A.; Morón, M.A., Riordan matrices in the reciprocation of quadratic polynomials, Linear algebra appl., 430, 2254-2270, (2009) · Zbl 1175.41029 [11] 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 [12] Rogers, D.G., Pascal triangles, Catalan numbers and renewal arrays, Discrete math., 22, 301-310, (1978) · Zbl 0398.05007 [13] Roman, Steven, The umbral calculus, (1984), Academic Press Inc · Zbl 0536.33001 [14] Roman, S.; Rota, G.-C., The umbral calculus, Adv. math., 27, 95-188, (1978) · Zbl 0375.05007 [15] Rota, G.C.; Kahaner, D.; Odlyzko, A., On the fundations of combinatorial theory, VIII: finite operators calculus, J. math. anal. appl., 42, 684-760, (1973) · Zbl 0267.05004 [16] Shapiro, L.W.; Getu, S.; Woan, W.J.; Woodson, L., The Riordan group, Discrete appl. math., 34, 229-239, (1991) · Zbl 0754.05010 [17] Sprugnoli, R., Riordan arrays and combinatorial sums, Discrete math., 132, 267-290, (1994) · Zbl 0814.05003 [18] 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.