An algebraic approach to Sheffer polynomial sequences

Abstract

A matrix approach to Sheffer polynomial sequences is proposed; in particular, two different determinantal forms of Sheffer sequences are given, the one as the function of a polynomial sequence of binomial type and the other as the function of the canonical base x i . The equivalence with the classical definitions of Sheffer and Roman and Rota is proven. Then, elementary matrix algebra tools are employed to reveal the known and unknown properties of Sheffer polynomials. Finally, classical and non-classical examples are also considered.

Determinantal approach to certain mixed special polynomials related to Gould–Hopper polynomials

“…Recently, determinantal forms of the Appell and Sheffer sequences are studied by Costabile and Longo in [4,5] respectively. We recall the following definition [5, p.4]:…”

“…The determinantal approach considered in [4,5] provides motivation to consider the determinantal forms of the new families of special polynomials.…”

Differential Equations for Classical and Non-Classical Polynomial Sets: A Survey

“…It is worth noting that the differential equations for Sheffer polynomial sets have been studied even with different methods (see [10][11][12][13]), but here we use only elements directly connected with the theory of polynomials.…”

Extended Laguerre–Appell polynomials via fractional operators and their determinant forms

“…The equivalence of the determinant approach with other existing approaches is also established. The simplicity of the algebraic approach to the Appell and Sheffer polynomials established in [6,7] allows several applications. The abovementioned research works by Costabile and Longo and the importance of operational methods in the theory of special functions motivated the authors to establish the determinant forms of the extended Laguerre-Appell polynomials.…”

“…An important approach based on a determinant definition was given in [5]. This approach is further extended to provide determinant definitions of the Appell and Sheffer polynomial sequences by Costabile and Longo in [6] and [7], respectively. The equivalence of the determinant approach with other existing approaches is also established.…”