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

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.

v5: minor corrections; remark added; to appear in J. Approx. Theory

References in corpus (2)

• Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems
• LU-decomposition of a noncommutative linear system and Jacobi polynomials

Cited by in corpus (7)

• Matrix-valued orthogonal polynomials related to the quantum analogue of $(SU(2)\times SU(2),\text{diag})$
• Matrix-valued Gegenbauer polynomials
• Well-Conditioned Fractional Collocation Methods Using Fractional Birkhoff Interpolation Basis
• Dual addition formulas: the case of continuous $q$-ultraspherical and $q$-Hermite polynomials
• A partial-sum deformation for a family of orthogonal polynomials
• Explicit matrix inverses for lower triangular matrices with entries involving continuous q-ultraspherical polynomials
• An evolution of matrix-valued orthogonal polynomials