A unified matrix approach to the representation of Appell polynomials

Abstract

In this paper we propose a unified approach to matrix representations of different types of Appell polynomials. This approach is based on the creation matrix -a special matrix which has only the natural numbers as entries and is closely related to the well known Pascal matrix. By this means we stress the arithmetical origins of Appell polynomials. The approach also allows to derive, in a simplified way, the properties of Appell polynomials by using only matrix operations.

Wronskian Appell polynomials and symmetric functions

“…For more examples and properties of Appell polynomials, we refer to [1] for a matrix approach or [30] for a probabilistic approach. As discussed in the introduction, we define the Wronskian Appell polynomial associated to a partition λ of length r and a given Appell sequence (A n ) ∞ n=0 to be…”

Shifted Generalized Pascal Matrices in the Context of Clifford Algebra-Valued Polynomial Sequences

“…Finally we emphasize that in this paper the key point was the role of both creation and shift matrices. In fact, we showed the relevance of those matrices in the representation of general hypercomplex sequences unifying and generalizing the approach already considered in [2] and [3].…”

“…. , m, has been highlighted in [2] as the main tool for the matrix representation of Appell polynomial sequences of one real variable and its extension to the representation of Sheffer sequences (cf. [1]).…”

Pascal Matrix Representation of Evolution of Polynomials

“…For almost all classical polynomials defined in ordinary way, as well, in a general way, the corresponding generating functions are known [9]. The generating functions usually are given in a form that reveals their property of being Appell polynomials due to the inclusion of the exponential function [3].…”

“…This is reminiscent of the quantal lowering operator. In literature, this matrix is called as creation matrix due to its role in construction of different types of Appell polynomials [3]. In this work, we investigate evolution of coefficients of the polynomial under simultaneous translations-shifts of the set of roots.…”