Eulerian polynomials as moments, via exponential Riordan arrays

TL;DR

This paper shows that certain Eulerian polynomials can be represented as moments of orthogonal polynomial families, using exponential Riordan arrays, and explores their generating functions and Hankel transforms.

Contribution

It introduces a novel connection between Eulerian polynomials and orthogonal polynomials via exponential Riordan arrays, providing new characterizations and generating functions.

Findings

Eulerian polynomials are moment sequences for orthogonal polynomials

Generated functions expressed as continued fractions

Hankel transforms of the polynomial sequences calculated

Abstract

Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the "descending power" Eulerian polynomials, and their once shifted sequence, are moment sequences for simple families of orthogonal polynomials, which we characterize in terms of their three-term recurrence. We obtain the generating functions of the polynomial sequences in terms of continued fractions, and we also calculate their Hankel transforms.