General Bivariate Appell Polynomials via Matrix Calculus and Related Interpolation Hints

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.

Bivariate general Appell interpolation problem

“…Remark 1 Differential relation (2a) is known in the literature (see, for example, [1,11,16] and references therein), but in different contexts and with different approaches.…”

“…The determinant forms for r n (x, y) and p n (x, y) [11], respectively, are fundamental in the sequel:…”

“…Theorem 1 (The main theorem) [11] For every f ∈ X the bivariate polynomial of total degree n given by…”

“…It is known [11] that in this case the elementary Appell sequence is {p n } b n∈IN , with p n (x, y) = H (2) n (x, y)…”

“…In this paper we will consider the bivariate interpolation problem proposed in [11]. We will give the unique solution expressed in the basis of the so-called bivariate, general Appell polynomial sequences [21].…”