On the Hankel transform of C-fractions

TL;DR

This paper investigates the relationship between C-fractions and the Hankel transforms of sequences, providing a closed-form formula and a method to differentiate transforms via multiplicity.

Contribution

It introduces a closed-form formula for the Hankel transform of sequences represented by C-fractions and defines multiplicity to distinguish between different transforms.

Findings

Derived a closed formula for Hankel transforms of C-fraction sequences

Linked the index of non-zero Hankel transform terms to C-fraction exponents

Introduced the concept of multiplicity to differentiate Hankel transforms

Abstract

We study the Hankel transforms of sequences whose generating function can be expressed as a C-fraction. In particular, we relate the index sequence of the non-zero terms of the Hankel transform to the powers appearing in the monomials defining the C-fraction. A closed formula for the Hankel transforms studied is given. As every power-series can be represented by a C-fraction, this gives in theory a closed form formula for the Hankel transform of any sequence. The notion of multiplicity is introduced to differentiate between Hankel transforms.