Recurrence relations of the multi-indexed orthogonal polynomials

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.