Lacunary Generating Functions for the Laguerre Polynomials

TL;DR

This paper applies umbral symbolic methods to derive new closed-form lacunary generating functions for Laguerre polynomials, revealing links to new special functions and extending previous quasi-monomial formulations.

Contribution

It introduces a novel umbral approach to lacunary generating functions for Laguerre polynomials, producing new closed-form expressions and connecting to a new family of special functions.

Findings

New closed-form lacunary generating functions for Laguerre polynomials

Connections to a new family of special functions

Extension of quasi-monomial theory

Abstract

Symbolic methods of umbral nature play an important and increasing role in the theory of special functions and in related fields like combinatorics. We discuss an application of these methods to the theory of lacunary generating functions for the Laguerre polynomials for which we give a number of new closed form expressions. We present furthermore the different possibilities offered by the method we have developed, with particular emphasis on their link to a new family of special functions and with previous formulations, associated with the theory of quasi monomials.