Recurrence Relations of the Multi-Indexed Orthogonal Polynomials

TL;DR

This paper investigates the recurrence relations of multi-indexed orthogonal polynomials, revealing they satisfy 3+2M term relations, extending classical cases and including Laguerre, Jacobi, Wilson, and Askey-Wilson types.

Contribution

It establishes the explicit form of recurrence relations for multi-indexed orthogonal polynomials, generalizing classical three-term relations to 3+2M terms.

Findings

Multi-indexed orthogonal polynomials satisfy 3+2M term recurrence relations.

Includes classical polynomials as the M=0 case.

Provides initial data for the lowest M+1 members.

Abstract

Ordinary orthogonal polynomials are uniquely characterized by the three term recurrence relations up to an overall multiplicative constant. We show that the newly discovered M-indexed orthogonal polynomials satisfy 3+2M term recurrence relations with non-trivial initial data of the lowest M+1 members. These include the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. The M=0 case is the corresponding classical orthogonal polynomials.