The Hankel transform of aerated sequences

摘要

This paper provides the connection between the Hankel transform and aerating transforms of a given integer sequence. Two different aerating transforms are introduced and closed-form expressions are derived for the Hankel transform of such aerated sequences. Combinations of both aerating and the Hankel transforms are also considered. Our results are general and can be applied to a wide class of integer sequences. As an application, we use our tools on the sequence of shifted Catalan numbers (C_(n+t))_(n∈?0). For that purpose, we need to evaluate the Hankel and Hankel-like determinants based on the Catalan numbers. Our approach is based on the results of Gessel and Viennot [Determinants, paths, and plane partitions preprint. Available from: http://www.cs.brandeis.edu/~ira] and more recent results of Krattenthaler [Determinants of (generalised) Catalan numbers. J. Statist. Plann. Inference 2010; 240: 2260-2270]. We generalize a sequence obtained by the series reversion of Q(x) = x/(1 + αx + βx~2) (studied in our previous paper [Boji?i? R, Petkovi? MD, Barry P. Hankel transform of a sequence obtained by series reversion. Integral Transforms Spec. Funct. DOI: 10.1080/10652469.2011.640326]) and provide the Hankel transform evaluation of that sequence and its shifted sequences.