mathematics Article General Bivariate Appell Polynomials via Matrix Calculus and Related Interpolation Hints Francesco Aldo Costabile, Maria Italia Gualtieri and Anna Napoli *   Citation: Costabile, F.A.; Gualtieri, M.I.; Napoli, A. General Bivariate Appell Polynomials via Matrix Calculus and Related Interpolation Hints. Mathematics 2021, 9, 964. https://doi.org/ 10.3390/math9090964 Academic Editor: Clemente Cesarano Received: 13 March 2021 Accepted: 23 April 2021 Published: 25 April 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department of Mathematics and Computer Science, University of Calabria, 87036 Rende (CS), Italy; francesco.costabile@unical.it (F.A.C.); mariaitalia.gualtieri@unical.it (M.I.G.) * Correspondence: anna.napoli@unical.it Abstract: An approach to general bivariate Appell polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations, determinant forms, differential equations and other properties. Some applications to linear functional and linear interpolation are sketched. New and known examples of bivariate Appell polynomial sequences are given. Keywords: Polynomial sequences; Appell polynomials; bivariate Appell sequence 1. Introduction Appell polynomials have many applications in various disciplines: probability the- ory [1–5], number theory [6], linear recurrence [7], general linear interpolation [8–12], operators approximation theory [13–17]. In [18], P. Appell introduced a class of polynomi- als by the following equivalent conditions: { A n } n∈IN is an Appell sequence ( A n being a polynomial of degree n) if either            dA n ( x) dx = nA n-1 ( x), n ≥ 1, A n (0)= α n , α 0 6 = 0, α n ∈ IR, n ≥ 0, A 0 ( x)= 1, or A(t)e xt = ∞ ∑ n=0 A n ( x) t n n! , where A(t)= ∞ ∑ k=0 α k t k k! , α 0 6 = 0, α k ∈ IR, k ≥ 0. Subsequentely, many other equivalent characterizations have been formulated. For example, in [19] [p. 87], there are seven equivalences. Properties of Appell sequences are naturally handled within the framework of modern classic umbral calculus (see [19,20] and references therein). Special polynomials in two variables are useful from the point of view of applications, particularly in probability [21], in physics, expansion of functions [22], etc. These poly- nomials allow the derivation of a number of useful identities in a fairly straightforward way and help in introducing new families of polynomials. For example, in [23] the au- thors introduced general classes of two variables Appell polynomials by using properties of an iterated isomorphism related to the Laguerre-type exponentials. In [24], the two- variable general polynomial (2VgP) family p n ( x, y) has been considered, whose members are defined by the generating function e xt φ(y, t)= ∞ ∑ n=0 p n ( x, y) t n n! , Mathematics 2021, 9, 964. https://doi.org/10.3390/math9090964 https://www.mdpi.com/journal/mathematics