Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays

TL;DR

This paper demonstrates how two combinatorial polynomials can be expressed as moments of orthogonal polynomial families using exponential Riordan arrays, enabling the derivation of their Hankel transforms.

Contribution

It introduces a novel method linking combinatorial polynomials to orthogonal polynomials via exponential Riordan arrays, providing a new approach to compute Hankel transforms.

Findings

Combinatorial polynomials can be represented as moments of orthogonal polynomials.

Hankel transforms of these polynomials can be explicitly derived.

Exponential Riordan arrays serve as effective tools for this analysis.

Abstract

In the case of two combinatorial polynomials, we show that they can exhibited as moments of paramaterized families of orthogonal polynomials, and hence derive their Hankel transforms. Exponential Riordan arrays are the main vehicles used for this.