Hankel transform of a sequence obtained by series reversion II - aerating transforms
Provides a theoretical tool for evaluating Hankel transforms of sequences, useful for mathematicians working on integer sequences and combinatorial identities.
This paper establishes a connection between the Hankel transform and aerating transforms of integer sequences, generalizing previous results to evaluate Hankel transforms of sequences involving Catalan numbers. The results are general and applicable to many other Hankel transform evaluations.
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 $\seqn{α^2 C_n-βC_{n+1}}$ and $\seqn{α^2 C_{n+1}-βC_{n+2}}$, where $C=\seqn{C_n}$ is the well-known sequence of Catalan numbers. This generalizes the results of Cvetkovi\' c, Rajković and Ivković \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.