Multiple orthogonal polynomials associated with the exponential integral

TL;DR

This paper introduces a new family of multiple orthogonal polynomials linked to exponential integral functions, providing explicit formulas, asymptotic zero distribution, and connections to random matrix theory.

Contribution

It develops explicit formulas, recurrence relations, and asymptotic analysis for a novel class of multiple orthogonal polynomials associated with exponential integral-based weights.

Findings

Explicit formulas for type I and II polynomials

Asymptotic zero distribution analysis

Connection to random matrix theory

Abstract

We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights $(w_1,w_2)$ on the positive real line, with $w_1(x)=x^\alpha e^{-x}$ the gamma density and $w_2(x) = x^\alpha E_{\nu+1}(x)$ a density related to the exponential integral $E_{\nu+1}$. We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider a related family of mixed type multiple orthogonal polynomials.