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Recurrence relations for polynomial sequences via Riordan matrices. (English) Zbl 1250.11029
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.

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)
Full Text: DOI
[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
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