Two-variable $-1$ Jacobi polynomials

TL;DR

This paper introduces a new two-variable generalization of Big -1 Jacobi polynomials, characterizes their properties, and derives their orthogonality measure and bispectral features through limiting processes and Pearson-type equations.

Contribution

It presents the first two-variable Big -1 Jacobi polynomials, detailing their construction, orthogonality, and bispectral properties, extending univariate polynomials to a bivariate setting.

Findings

Orthogonality measure obtained for the bivariate polynomials

Bispectral properties characterized via limiting process

Alternative derivation of weight function using Pearson equations

Abstract

A two-variable generalization of the Big $-1$ Jacobi polynomials is introduced and characterized. These bivariate polynomials are constructed as a coupled product of two univariate Big $-1$ Jacobi polynomials. Their orthogonality measure is obtained. Their bispectral properties (eigenvalue equations and recurrence relations) are determined through a limiting process from the two-variable Big $q$-Jacobi polynomials of Lewanowicz and Wo\'zny. An alternative derivation of the weight function using Pearson-type equations is presented.