Centrally symmetric orthogonal polynomials in two variables

Commun. Korean Math. Soc. 1997 Vol. 12, No. 3, 645-653

Jeong Keun Lee Sunmoon university

Abstract : We study centrally symmetric orthogonal polynomials satisfying an admissible partial differential equation of the form $$Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_x+Eu_y=\lambda_nu,$$ where $A,B,\cdots,E$ are polynomials independent of $n$ and $\lambda_n$ is the eigenvalue parameter depending on $n$. We show that they are either the product of Hermite polynomials or the circle polynomials up to a complex linear change of variables. Also we give some properties of them.

Keywords : orthogonal polynomials in two variables, second order partial differential equations, centrally symmetric, Hermite polynomials, circle po-lynomials