Inverse Series Relations And Classical Combinatorial Identities

By means of inverse series relations, this thesis will systematically investigate identi-ties for classical Catalan numbers, generalized hypergeometric series and basic hypergeo-metric series. The content is summarized as follows:1. The higher moments of Catalan numbers are evaluated by combining inverse series relations with symmetric functions and telescoping method. This resolves a problem recently proposed by Gutierrez et al (2007).2. Several identities and reciprocal relations are established for terminating hypergeo-metric series, which is fulfilled by investigating the dual relations to the formulae of Chu-Vandermonde-Gauss, Pfaff-Saalschutz and the binomial convolution due to Hagen and Rothe through the multiplicate form of Gould-Hsu inverse series relations.3. Thirty closed formulae for a large class of terminating hypergeometric 3F2(3/4) series are obtained by applying the original Gould-Hsu inversions to Pfaff-Saalschiitz summation theorem together with four series transformations and two contiguous relations.4. The inverse series relations due to Carlitz (1973) are utilized to derive several trans-formation formulae for nonterminating q-series, which are in turn employed to review systematically the identities of Rogers-Ramanujan type moduli 5,7,8,10,14 and 27.