Recurrence relations for the Sheffer sequences

Abstract

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.

Riordan arrays and d-orthogonality

“…This leads to the important corollary which is an analogue of [35] Corollary 9. If L = (g(x), f (x)) is a Riordan array and P = S L is (d + 2)-banded lower matrix given by (18), then L −1 is the coefficient array of the family of d-OPS.…”

“…In this case, the corresponding Stieltjes matrix takes the following form . Therefore, from (35) and (36) we obtain f (x) = b + xe τx and…”

“…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 determinantal approach to Sheffer sequences

“…It should be noted that very recently, another Yang [24] established a different determinantal representation for Sheffer sequences by using an approach based on the production matrices of Riordan arrays. Now, let us introduce some definitions and results briefly.…”

An algebraic approach to Sheffer polynomial sequences

“…In [9][10][11], there are two different algebraic approaches to Appell polynomials [12] that are a particular case of Sheffer polynomials. Recently, in [13] a determinantal form for Sheffer polynomials has been proposed. In this paper, we offer a new algebraic approach to Sheffer polynomial sequences, extending the one in [10] for Appell polynomials.…”