Moment Representations of the Exceptional $X_1$-Laguerre Orthogonal Polynomials

TL;DR

This paper introduces new moment-based representations for Type I $X_1$-Laguerre exceptional orthogonal polynomials, including recursive formulas and generating functions, enhancing understanding of their structure.

Contribution

It provides two novel moment representations for the $X_1$-Laguerre polynomials, including a more elegant formula using adjusted moments and their recursive relations.

Findings

Derived two moment representations for $X_1$-Laguerre polynomials.

Introduced adjusted moments and their generating function.

Observed a detachedness of the first two moments from the others.

Abstract

Exceptional orthogonal Laguerre polynomials can be viewed as an extension of the classical Laguerre polynomials per excluding polynomials of certain order(s) from being eigenfunctions for the corresponding exceptional differential operator. We are interested in the (so-called) Type I $X_1$-Laguerre polynomial sequence $\{L_n^\alpha\}_{n=1}^\infty$, $\text{deg} \,p_n = n$ and $\alpha>0$, where the constant polynomial is omitted. We derive two representations for the polynomials in terms of moments by using determinants. The first representation in terms of the canonical moments is rather cumbersome. We introduce adjusted moments and find a second, more elegant formula. We deduce a recursion formula for the moments and the adjusted ones. The adjusted moments are also expressed via a generating function. We observe a certain detachedness of the first two moments from the others.