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

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, recent...

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Bibliographic Details
Published in:arXiv.org 2011-12
Main Authors: Bojičić, Radica, Petković, Marko D, Barry, Paul
Format: Article
Language:English
Subjects:
Aeration
Rational functions
Reversion
Sequences
Series (mathematics)
Online Access:Get full text
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Summary: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, 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.
ISSN:2331-8422