A new family of orthogonal polynomials in three variables

TL;DR

This paper introduces a six-parameter family of orthogonal polynomials in three variables, explores their properties, recurrence relations, differential equations, and connections to existing polynomials, expanding the mathematical framework for multivariate orthogonal polynomials.

Contribution

It presents a novel six-parameter generalization of three-variable orthogonal polynomials on the simplex with detailed properties and relations.

Findings

Derived sparse recurrence relations for the new polynomials

Established second order partial differential equations

Connected the new polynomials to existing four-parameter polynomials

Abstract

In this paper we introduce a six-parameter generalization of the four-parameter three-variable polynomials on the simplex and we investigate the properties of these polynomials. Sparse recurrence relations are derived by using ladder relations for shifted univariate Jacobi polynomials and bivariate polynomials on the triangle. Via these sparse recurrence relations, second order partial differential equations are presented. Furthermore, some connection relations are obtained between these polynomials. Finally, new results for the four-parameter three-variable polynomials on the simplex are given.