An orthogonal polynomial coefficient formula for the Hankel transform

TL;DR

This paper presents an explicit formula connecting the Hankel transform of regular sequences with orthogonal polynomial coefficients, enabling new combinatorial identities and structural insights.

Contribution

It introduces a novel explicit formula linking Hankel transforms to orthogonal polynomial coefficients for regular sequences.

Findings

Derived new combinatorial identities using the formula

Gained structural insights into the Hankel transform of sequences

Applied the formula to sequences of combinatorial interest

Abstract

We give an explicit formula for the Hankel transform of a regular sequence in terms of the coefficients of the associated orthogonal polynomials and the sequence itself. We apply this formula to some sequences of combinatorial interest, deriving interesting combinatorial identities by this means. Further insight is also gained into the structure of the Hankel transform of a sequence.