Moment Representations of Type I X2 Exceptional Laguerre Polynomials

TL;DR

This paper introduces a specific flag for X_m exceptional Laguerre polynomials, derives their determinantal representations, and establishes a recursion for associated moments, advancing understanding of their structure beyond classical polynomials.

Contribution

The paper presents a new canonical flag for X_m exceptional Laguerre polynomials and derives their determinantal representations with a recursion for adjusted moments.

Findings

Derived determinantal representations for X_2 exceptional Laguerre polynomials.

Established a recursion formula for the adjusted moments of the exceptional weights.

Identified a specific flag that simplifies the representation and moment calculations.

Abstract

The $X_m$ exceptional orthogonal polynomials (XOP) form a complete set of eigenpolynomials to a differential equation. Despite being complete, the XOP set does not contain polynomials of every degree. Thereby, the XOP escape the Bochner classification theorem. In literature two ways to obtain XOP have been presented. When m=1, Gram-Schmidt orthogonalization of a so-called "flag" was used. For general m, the Darboux transform was applied. Here, we present a possible flag for the X_m exceptional Laguerre polynomials. We can write more about this. We only want to make specific picks when we also derive determinantal representations. There is a large degree of freedom in doing so. Further, we derive determinantal representations of the X_2 exceptional Laguerre polynomials involving certain adjusted moments of the exceptional weights. We find a recursion formula for these adjusted moments. The particular canonical flag we pick keeps both the determinantal representation and the moment recursion manageable.