mathematics Article General Bivariate Appell Polynomials via Matrix Calculus and Related Interpolation Hints Francesco Aldo Costabile, Maria Italia Gualtieri and Anna Napoli * Department of Mathematics and Computer Science, University of Calabria, 87036 Rende (CS), Italy; [email protected] (F.A.C.); [email protected] (M.I.G.) * Correspondence: [email protected] 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: fAngn2IN is an Appell sequence (An being a Citation: Costabile, F.A.; polynomial of degree n) if either Gualtieri, M.I.; Napoli, A. General 8 Bivariate Appell Polynomials via d An(x) > = nA − (x), n ≥ 1, Matrix Calculus and Related > n 1 <> dx Interpolation Hints. Mathematics 2021, A (0) = a , a 6= 0, a 2 IR, n ≥ 0, 9, 964. https://doi.org/ > n n 0 n > 10.3390/math9090964 :> A0(x) = 1, Academic Editor: Clemente Cesarano or ¥ n xt t A(t)e = ∑ An(x) , Received: 13 March 2021 n=0 n! Accepted: 23 April 2021 ¥ tk Published: 25 April 2021 where A(t) = ∑ ak , a0 6= 0, ak 2 IR, k ≥ 0. k=0 k! Publisher’s Note: MDPI stays neutral Subsequentely, many other equivalent characterizations have been formulated. For with regard to jurisdictional claims in example, in [19] [p. 87], there are seven equivalences. published maps and institutional affil- Properties of Appell sequences are naturally handled within the framework of modern iations. 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 Copyright: © 2021 by the authors. way and help in introducing new families of polynomials. For example, in [23] the au- Licensee MDPI, Basel, Switzerland. thors introduced general classes of two variables Appell polynomials by using properties This article is an open access article of an iterated isomorphism related to the Laguerre-type exponentials. In [24], the two- distributed under the terms and variable general polynomial (2VgP) family pn(x, y) has been considered, whose members conditions of the Creative Commons are defined by the generating function Attribution (CC BY) license (https:// ¥ n creativecommons.org/licenses/by/ xt t e f(y, t) = ∑ pn(x, y) , 4.0/). n=0 n! Mathematics 2021, 9, 964. https://doi.org/10.3390/math9090964 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 964 2 of 29 ¥ tk where f(y, t) = ∑ fk(y) . k=0 k! Later, the authors considered the two-variable general Appell polynomials (2VgAP) ( ) f gb denoted by p An x, y based on the sequence pn n2IN, that is ¥ n xt t A(t)e f(y, t) = ∑ p An(x, y) , n=0 n! ¥ tk where A(t) = ∑ ak , a0 6= 0, ak 2 IR, k ≥ 0. k=0 k! These polynomials are framed within the context of monomiality principle [24–27]. Generalizations of Appell polynomials can be also found in [22,28–31] (see also the references therein). In this paper, we will reconsider the 2VgAP, but with a systematic and alternative theory, that is matrix calculus-based. To the best of authors knowledge, a systematic approach to general bivariate Appell sequences does not appear in the literature. New properties are given and a general linear interpolation problem is hinted. Some applications of the previous theory are given and new families of bivariate polynomials are presented. Moreover a biorthogonal system of linear functionals and polynomials is constructed. In particular, the paper is organized as follows: in Section2 we give the definition and the first characterizations of general bivariate Appell polynomial sequences; in Sections3–5 we derive, respectively, matrix form, recurrence relations and determinant forms for the elements of a general bivariate Appell polynomial sequence. These sequences satisfy some interesting differential equations (Section6) and properties (Section7). In Section8 we consider the relations with linear functional of linear interpolation. Section9 introduces new and known examples of polynomial sequences. Finally, Section 10 contains some concluding remarks. We point out that the first recurrence formula and the determinant forms, as well as the relationship with linear functionals and linear interpolation, to the best of authors’ knowledge, do not appear in the literature. We will adopt the following notation for the derivatives of a polynomial f ¶i+j f f (i,j) = , f (0,0) = f (x, y), f (i,j)(a, b) = f (i,j)(x, y) . ¶xi¶yj (x,y)≡(a,b) A set of polynomials is denoted, for example, by fp0,..., pn j n 2 INg, where the sub- scripts 0, ... , n represent the (total) degree of each polynomial. Moreover, for polynomial f g f gb sequences, we will use the notation an n2IN for univariate sequence and rn n2IN in the bivariate case. Uppercase letters will be used for particular and well-known sequences. 2. Definition and First Characterizations Let A(t) be the power series ¥ tk A(t) = ∑ ak , a0 6= 0, ak 2 IR, k ≥ 0, (1) k=0 k! (usually a0 = 1) and let f(y, t) be the two-variable real function defined as ¥ tk f(y, t) = ∑ jk(y) , (2) k=0 k! where jk(y) are real polynomials in the variable y, with j0(y) = 1. Mathematics 2021, 9, 964 3 of 29 It is known ([19], p. 78) that the power series A(t) generates the univariate Appell polynomial sequence fAngn2IN such that n n k A0(x) = 1, An(x) = ∑ an−kx , n ≥ 1. (3) k=0 k Now we consider the bivariate polynomals rn with real variables. We denote by A(f, A), or simply A where there is no possibility of misunderstanding, the set of bivariate f gb polynomial sequences rn n2IN such that 8 r0(x, y) = 1 (4a) > > (1,0) < rn (x, y) = n rn−1(x, y), n ≥ 1 (4b) n > n > r (0, y) = a j (y). (4c) :> n ∑ n−k k k=0 k In the following, unless otherwise specified, the previous hypotheses and notations will always be used. f gb Remark 1. We observe that in [21,32] a polynomial sequence Pi n2IN is said to satisfy the Appell condition if ¶ P (t, x) = P − (t, x), P (t, x) = 1. ¶t i i 1 0 This sequence in [32] is used to obtain an expansion of bivariate, real functions with integral remainder (generalization of Sard formula [33]. Nothing is said about the theory of this kind of sequences. f gb A Proposition 1. A bivariate polynomial sequence rn n2IN is an element of if and only if n n rn(x, y) = ∑ An−k(x)jk(y), n ≥ 1. (5) k=0 k f gb 2 A Proof. If rn n2IN , relations (4a) hold. Then, by induction and partial integration with respect to the variable x ([19] p. 93), we get relation (5), according to (3). Vice versa, from (5), we easily get (4a). f gb A Proposition 2. A bivariate polynomial sequence rn n2IN is an element of if and only if ¥ n xt t A(t)e f(y, t) = ∑ rn(x, y) . (6) n=0 n! f gb 2 A Proof. If rn n2IN , from Proposition1 the identity (5) holds. Then ! ¥ tn ¥ n n tn ∑ rn(x, y) = ∑ ∑ An−k(x)jk(y) . n=0 n! n=0 k=0 k n! From the Cauchy product of series, according to (1) and (2), we get (6). Vice-versa, from (6) we obtain (5). Therefore relations (4a) hold. We call the function F(x, y; t) = A(t)extf(y, t) exponential generating function of the f gb bivariate polynomial sequence rn n2IN. Remark 2. From Propositions1 and2 we note explicitly that relations (4a) are equivalent to the identity (6). Mathematics 2021, 9, 964 4 of 29 ( ) = ( ) = ( ) ≡ > f gb Example 1. Let f y, t 1, that is j0 y 1, jk y 0, k 0. Then rn n2IN, constructed as in Proposition1, or, equivalently, Proposition2, is a polynomial sequence in one variable, with elements rn(x, y) ≡ rn(x) = An(x). f gb Therefore rn n2IN is a univariate Appell polynomial sequence [18,19]. Example1 suggests us the following definition. f gb 2 A Definition 1. A bivariate polynomial sequence rn n2IN , that is a polynomial sequence satisfying relations (4a) or relation (6), is called general bivariate Appell polynomial sequence. Remark 3. (Elementary general bivariate Appell polynomial sequences) Assuming A(t) = 1, that is a0 = 1, ai = 0, i ≥ 1, relations (4a) become 8 r (x, y) = 1 (7a) > 0 < (1,0) rn (x, y) = n rn−1(x, y), n > 1 (7b) :> rn(0, y) = jn(y). (7c) n Moreover, the univariate Appell sequence is An(x) = x , n ≥ 0. Hence, from (5), n n n−k rn(x, y) = ∑ x jk(y), n ≥ 1. (8) k=0 k Relation (6) becomes ¥ n xt t e f(y, t) = ∑ rn(x, y) . (9) n=0 n! f gb In this case, we call the polynomial sequence rn n2IN elementary bivariate Appell se- f gb quence.