Hankel transform of a sequence obtained by series reversion II - aerating transforms

TL;DR

This paper explores the relationship between Hankel transforms and aerating transforms of integer sequences, applying these to evaluate Hankel transforms of series reversion of rational functions and shifted Catalan sequences.

Contribution

It introduces new connections between Hankel and aerating transforms, providing novel evaluations of Hankel transforms for sequences related to Catalan numbers and their series reversion.

Findings

Hankel transforms of sequences involving Catalan numbers are explicitly evaluated.

New formulas for Hankel-like determinants with Catalan number entries are derived.

Results generalize previous work and can be applied to various Hankel transform problems.

Abstract

This paper provides the connection between the Hankel transform and aerating transforms of a given integer sequence. Results obtained are used to establish a completely different Hankel transform evaluation of the series reversion of a certain rational function $Q(x)$ and shifted sequences, recently published in our paper \cite{part1}. For that purpose, we needed to evaluate the Hankel transforms of the sequences $\seqn{\alpha^2 C_n-\beta C_{n+1}}$ and $\seqn{\alpha^2 C_{n+1}-\beta C_{n+2}}$, where $C=\seqn{C_n}$ is the well-known sequence of Catalan numbers. This generalizes the results of Cvetkovi\' c, Rajkovi\'c and Ivkovi\'c \cite{CRI}. Also, we need the evaluation of Hankel-like determinants whose entries are Catalan numbers $C_n$ and which is based on the recent results of Krattenthaler \cite{krattCat}. The results obtained are general and can be applied to many other Hankel transform evaluations.