Recurrence relations for polynomial sequences via Riordan matrices

Abstract

We give recurrence relations for any family of generalized Appell polynomials unifying so some known recurrences of 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 families of polynomials. 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.

Riordan arrays and d-orthogonality

“…Conversely, starting from the sequences defined by (22), the infinite array (a n,k ) n,k≥0 defined by (27) is an exponential Riordan array.…”

“…They showed that some classical Riordan arrays have triple factorizations of the form P = P CF , where P , C, F are also Riordan arrays. We notice also that the inverse of the Stieltjes matrix represents a coefficient of the sequence polynomials [3,2,22,35]. In [34], the authors present new factorizations, and provide extensive examples of families of classical polynomials.…”

A symbolic treatment of Riordan arrays

“…Hence, after applying the evaluation map E, formula (14) gives the summation formula n k=0 r n,k f k = n! [t n ] d(t)f (h(t)) .…”

Recurrence relations for the Sheffer sequences

“…In [7], Luzón introduced a new notation T(f |g) to represent the Riordan arrays and gave a recurrence relation for the polynomial sequences associated to Riordan arrays. In this paper, using the production matrix of an exponential Riordan array, we give a recurrence relation for the Sheffer polynomial sequence.…”