Recurrence relations for polynomial sequences via Riordan matrices

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.

[1]  Donald E. Knuth.  Convolution polynomials , 1992 .

[2]  Riordan matrices in the reciprocation of quadratic polynomials , 2009 .

[3]  Ralph P. Boas,et al.  Polynomial Expansions of Analytic Functions , 1958 .

[4]  Hana Kim,et al.  Simple proofs of open problems about the structure of involutions in the Riordan group , 2008 .

[5]  Gian-Carlo Rota,et al.  On the foundations of combinatorial theory. VIII. Finite operator calculus , 1973 .

[7]  D. G. Rogers,et al.  Pascal triangles, Catalan numbers and renewal arrays , 1978, Discret. Math..

[8]  Tianming Wang,et al.  Generalized Riordan arrays , 2008, Discret. Math..

[9]  Hana Kim,et al.  Riordan group involutions , 2008 .

[10]  Ana Luzón,et al.  Ultrametrics, Banach's fixed point theorem and the Riordan group , 2008, Discret. Appl. Math..

[11]  G. Rota.  On the Foundations of Combinatorial Theory , 2009 .

[12]  Leetsch C. Hsu,et al.  The Sheffer group and the Riordan group , 2007, Discret. Appl. Math..

[13]  Mourad E. H. Ismail,et al.  A -umbral calculus , 1981 .

[14]  Asamoah Nkwanta,et al.  On Some (Pseudo) Involutions in the Riordan Group , 2005 .

[15]  Renzo Sprugnoli,et al.  Riordan arrays and combinatorial sums , 1994, Discret. Math..

[16]  Renzo Sprugnoli,et al.  On Some Alternative Characterizations of Riordan Arrays , 1997, Canadian Journal of Mathematics.

[17]  Lou Shapiro.  Some Open Questions about Random Walks, Involutions, Limiting Distributions, and Generating Functions , 2001, Adv. Appl. Math..

[18]  M. Amlouk,et al.  Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition , 2007 .

[19]  Louis W. Shapiro,et al.  The Riordan group , 1991, Discret. Appl. Math..

[20]  E. Bell,et al.  The History of Blissard's Symbolic Method, With a Sketch of its Inventor's Life , 1938 .

[21]  I-Chiau Huang.  Inverse Relations and Schauder Bases , 2002, J. Comb. Theory, Ser. A.