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

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.

19 pages

References in corpus (5)

• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Solvable rational extensions of the isotonic oscillator
• Exceptional orthogonal polynomials and the Darboux transformation
• Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials
• A Spectral Study of the Second-Order Exceptional $X_1$-Jacobi Differential Expression and a Related Non-classical Jacobi Differential Expression

Cited by in corpus (2)

• Moment Representations of Exceptional $X_1$ Orthogonal Polynomials
• Determinantal Formulas for Exceptional Orthogonal Polynomials