Explicit matrix inverses for lower triangular matrices with entries involving Jacobi polynomials

TL;DR

This paper derives explicit inverses for a family of lower triangular matrices involving Jacobi polynomials, simplifies the formulas in special cases, and connects these matrices to group structures and orthogonal polynomial relations.

Contribution

It provides explicit inverse formulas for matrices with Jacobi polynomial entries and links them to group generators and orthogonal polynomial connection relations.

Findings

Explicit inverse formulas involving sums of Jacobi polynomials

Simplification of formulas in the Gegenbauer case

Connection to group structures and orthogonal polynomial relations

Abstract

For a two-parameter family of lower triangular matrices with entries involving Jacobi polynomials an explicit inverse is given, with entries involving a sum of two Jacobi polynomials. The formula simplifies in the Gegenbauer case and then one choice of the parameter solves an open problem in a recent paper by Koelink, van Pruijssen & Roman. The two-parameter family is closely related to two two-parameter groups of lower triangular matrices, of which we also give the explicit generators. Another family of pairs of mutually inverse lower triangular matrices with entries involving Jacobi polynomials, unrelated to the family just mentioned, was given by J. Koekoek & R. Koekoek (1999). We show that this last family is a limit case of a pair of connection relations between Askey-Wilson polynomials having one of their four parameter in common.