A matrix Rodrigues formula for classical orthogonal polynomials in two variables

Abstract

Classical orthogonal polynomials in one variable can be characterized as the only orthogonal polynomials satisfying a Rodrigues formula. In this paper, using the second kind Kronecker power of a matrix, a Rodrigues formula is introduced for classical orthogonal polynomials in two variables.

On differential properties for bivariate orthogonal polynomials

“…In this work, we will make an intensive use of the symmetrized second kind Kronecker power (see [3], p. 236) of a symmetric and square matrix A = (a i, j ) 1 i, j=0 of dimension 2. Consider the linear transformation defined by means of z = A t, where t = (t 1 , t 2 ) t , and z = (z 1 , z 2 ) t .…”

“…These operators can be extended for higher order derivatives [1], and for matrices [5,6]. In fact, if we denote…”

“…Remark 4.11 Matrices [h] are also connected with a matrix Rodrigues formula for a WOPS associated with a moment functional u satisfying (15). We kindly suggest the reader check reference [1].…”

“…The author showed that this property holds if and only if a y = c x = 0, and called this class of partial differential equations the basic class. However, A. S. Lyskova did not discuss the orthogonality of partial derivatives of orthogonal polynomials satisfying the partial differential equation (1).…”

“…In [14], Lee et al classify, up to a real change of variables, all partial differential equations (1) which are in the basic class, and show that partial derivatives of any order of orthogonal polynomial solutions to the partial differential equations in the basic class are also orthogonal.…”