Discrete orthogonality relations for multi-indexed Laguerre and Jacobi 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 a more encompassing characterization of orthogonal polynomials than the three term recurrence relations. As the multi-indexed orthogonal polynomials start at a positive degree ℓD≥1, the three term recurrence relations are broken. The extra ℓD “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 ℓD+N. 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.