Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types

TL;DR

This paper derives new recurrence relations for multi-indexed Meixner-Pollaczek and continuous Hahn orthogonal polynomials, and explores their implications for quantum systems, extending previous work on other polynomial types.

Contribution

It introduces novel recurrence relations and quantum operator structures for these specific multi-indexed orthogonal polynomials, expanding the mathematical framework.

Findings

Derived $3+2M$ term recurrence relations with variable coefficients.

Established $1+2L$ term recurrence relations with constant coefficients.

Connected these relations to quantum mechanical creation and annihilation operators.

Abstract

In previous papers, we discussed the recurrence relations of the multi-indexed orthogonal polynomials of the Laguerre, Jacobi, Wilson, Askey-Wilson, Racah and $q$-Racah types. In this paper we explore those of the Meixner-Pollaczek and continuous Hahn types. For the $M$-indexed Meixner-Pollaczek and continuous Hahn polynomials, we present $3+2M$ term recurrence relations with variable dependent coefficients and $1+2L$ term ($L\geq M+1$) recurrence relations with constant coefficients. Based on the latter, the generalized closure relations and the creation/annihilation operators of the quantum mechanical systems described by the multi-indexed Meixner-Pollaczek and continuous Hahn polynomials are obtained.