Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials

TL;DR

This paper derives simplified expressions for multi-indexed Laguerre and Jacobi polynomials, which are orthogonal polynomials with complex degree structures, facilitating their application in mathematical physics.

Contribution

It introduces two new simplified forms of these polynomials based on identities, enhancing their usability and understanding.

Findings

Derived simplified expressions for multi-indexed Laguerre and Jacobi polynomials.

Established parity transformation properties of multi-indexed Jacobi polynomials.

Provided explicit Wronskian-based formulas for these polynomials.

Abstract

The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials is derived based on that of the Jacobi polynomial.