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

Leandro Cagliero ^[1] ; Tom H. Koornwinder ^[2]
1. [1] Universidad Nacional de Córdoba
  
  Universidad Nacional de Córdoba
  
  Argentina
2. [2] University of Amsterdam
  
  University of Amsterdam
  
  Países Bajos
Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 193, Nº 1 (May 2015), 2015, págs. 20-38
Idioma: inglés
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Resumen
- 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 & Román. 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 parameters in common.