Applications Of Inversion Techniques In Combinatorial Identities

This thesis establishes several combinatorial identities and generalizes Stirling numbers of the second kind. The contents is as follows:1. The first chapter introduces the phylogeny of combinatorial identities, especially hypergeometric series identities, and Stirling numbers of the senond kind.2. In the second chapter, by means of Legendre inversions, we prove two terminating hypergeometric series identities. Their reversals and linear combinations yield more results. Besides, by the property of polynomials we establish the inherent relations of several identities.3. In 2000. Chu Wenchang established duplicate inverse series relations and derived numerous hypereometric series identities. As further applications, we estalish several new hypergeometric formulas, find sveral Hagen-Rothe type transforms and deduce a class of Fox-Wright function identities.4. Based on partitions of finite sets and restrictions to the distances among the elements in each part, we generalize Stirling numbers of the second kind in two different ways. The recurrence relations, generating functions and explicit formulae are established consequently. Some combinatorial interpretations are also provided.