Combinatorial properties of irreducible Laguerre polynomials in two variables

TL;DR

This paper explores the properties of two-variable irreducible Laguerre polynomials, deriving recurrence relations, generating functions, and connections to standard polynomials, expanding understanding of their combinatorial structure.

Contribution

It introduces new recurrence formulas and generating functions for two-variable irreducible Laguerre polynomials, linking them to classical polynomials.

Findings

Derived recurrence formulas for the polynomials

Established a generating function for the polynomials

Connected sums of these polynomials to standard polynomial families

Abstract

Following our earlier work, where doubly indexed and irreducible over Q two-variable Laguerre polynomials were introduced, we prove for such polynomials some recurrence formulas and obtain a generating function. In addition, we show how certain sums of such polynomials with a fixed total degree relate to some standard polynomials.