Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials

TL;DR

This paper extends discrete orthogonality relations to multi-indexed Laguerre and Jacobi polynomials, revealing new orthogonal structures and identifying necessary lower degree polynomials and Christoffel numbers through differential operator analysis.

Contribution

It demonstrates that discrete orthogonality relations hold for multi-indexed Laguerre and Jacobi polynomials, despite broken three-term recurrence relations, by analyzing differential operators around polynomial zeros.

Findings

Orthogonality relations hold for multi-indexed Laguerre and Jacobi polynomials.

Identification of extra lower degree polynomials needed for orthogonality.

Determination of Christoffel numbers for these polynomials.

Abstract

The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also hold for multi-indexed Laguerre and Jacobi polynomials, which are new orthogonal polynomials obtained by deforming these classical orthogonal polynomials. The discrete orthogonality relations could be considered as more encompassing characterisation of orthogonal polynomials than the three term recurrence relations. As the multi-indexed orthogonal polynomials start at a positive degree $\ell_{\mathcal D}\ge1$, the three term recurrence relations are broken. The extra $\ell_{\mathcal D}$ `lower degree polynomials', which are necessary for the discrete orthogonality relations, are identified. The corresponding Christoffel numbers are determined. The main results are obtained by the blow-up analysis of the second order differential operators governing the multi-indexed orthogonal polynomials around the zeros of these polynomials at a degree $\ell_{\mathcal D}+\mathcal{N}$. %changed The discrete orthogonality relations are shown to hold for another group of `new' orthogonal polynomials called Krein-Adler polynomials based on the Hermite, Laguerre and Jacobi polynomials.