Bivariate second-order linear partial differential equations and orthogonal polynomial solutions

Abstract

In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second-order linear partial differential equations, which are admissible potentially self-adjoint and of hypergeometric type. General formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation, using vector matrix notation. Moreover, Rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given. Finally, as illustration, these results are applied to specific Appell and Koornwinder orthogonal polynomials, solutions of the same partial differential equation.

Linear Partial Divided-Difference Equation Satisfied by Multivariate Orthogonal Polynomials on Quadratic Lattices

“…We give explicitly the recurrences satisfied by those families of bivariate Racah polynomials defined in (6) and (15). The family of monic bivariate Racah polynomials is introduced from the three-term recurrence relations it obeys, by following a similar approach as already considered in the continuous case [4], discrete case [3] and their q-analogues [1]. Moreover, by using these results we explicitly solve the connection problem between bivariate Racah polynomials (6) and (15).…”

“…To illustrate the truthful of our conjectures presented in sections 2.1 and 4.2, we consider the case p = 3 for the trivariate continuous Hahn polynomials defined by [20] H n,m,r (a 1 , e 2 , e 3 , a 4…”

Recurrence Relations for Orthogonal Polynomials on Triangular Domains

“…Orthogonal polynomials on the interior of the triangle based on second-order linear partial differential equations is considered in [7]. The approach is different when considering the boundary of the triangle which needs applying the directional derivatives.…”

Linear partial q-difference equations on q-linear lattices and their bivariate q-orthogonal polynomial solutions

“…where f k (x) = f k+1 (qx) ; (4) If P n (x) = a nk x k then a nk /a n,k−1 is a rational function of q n and q k ; (5) The moments associated with {P n (x)} satisfy (1.2) M n = a + bq n c + dq n M n−1 , a d − b c = 0 , where M n are either the power moments (moments against x k ) or the generalized moments (moments against the q-shifted factorial (x; q) k = k−1 j=0 (1 − xq j )). Hahn's investigation led him to the most general set of polynomials belonging to this class [8], now called q-Hahn polynomials:…”

“…In more recent papers the second-order linear partial differential equations of the hypergeometric type [5] and their discretization on uniform lattices [4,6,44,45], as well as a general way of introducing orthogonal polynomial families in two discrete variables on the simplex [43], have been analyzed. Therefore, it is possible to generalize the univariate classical orthogonal polynomials to the bivariate and multivariate versions by requiring that they obey a second-order partial differential equation of the hypergeometric type (continuous case) [32,48], or a second-order partial difference equation of the hypergeometric type (discrete case), as indicated before.…”