Multiple Laguerre polynomials: Combinatorial model and Stieltjes moment representation

TL;DR

This paper provides a combinatorial interpretation and an integral representation of multiple Laguerre polynomials of the first kind, revealing their structure as multidimensional Stieltjes moment sequences for non-positive x.

Contribution

It generalizes the digraph model for Laguerre polynomials to multiple Laguerre polynomials and offers an explicit integral representation.

Findings

Combinatorial interpretation via digraph model

Explicit integral representation for the polynomials

Identification as multidimensional Stieltjes moment sequence for x ≤ 0

Abstract

I give a combinatorial interpretation of the multiple Laguerre polynomials of the first kind of type II, generalizing the digraph model found by Foata and Strehl for the ordinary Laguerre polynomials. I also give an explicit integral representation for these polynomials, which shows that they form a multidimensional Stieltjes moment sequence whenever $x \le 0$.