Orthogonal polynomials in two variables

Abstract

Some properties of orthogon~l (and generalized orthogonal) polynomial sets in two variables are obtained, in particular a characterization of such sets based on generating functions. Then those li+~ear homogeneous partial differential eqnations o[ the form Lily]-bkvo:O, having a set of polynomials as soluhon, are characterized; and a detailed study is made of all such equations of second order ~vhose polynomial solutions form an orthogonal (or generalized orthogonal) set. 1. -Introduction.

Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions

“…12 Each eigenvalue is multiply degenerate. However, each separable coordinate system gives rise to an orthonormal basis of polynomial eigenfunctions in this space and breaks the degeneracy.…”

Bivariate version of the Hahn-Sonine theorem

“…The condition that ∂ x P 0n = 0 for all n ≥ 0 is not too restrictive. In fact, all weak orthogonal polynomials found by Krall and Sheffer [11] satisfy this condition.…”

“…, there is a unique moment functional σ, which is called the canonical moment functional of {Φ n } ∞ n=0 , defined by the conditions [11]). For any moment functional σ, the following statements are equivalent : …”

“…Then it is well known (see [6], [11]) that σ satisfies the following functional equations, called the moment equations,…”

“…The important work in this direction was done by Krall and Sheffer in 1967 [11]. But there have been only a few results in this area since Krall and Sheffer.…”