Proofs And Applications Of Some Combinatorical Identities

By means of analytic method and technique, such as functional equations congruence, mechanized method, this dissertation investigates the proofs and applications of several combinatorial identities. Giving new proofs of known identities, the author derives several new and meaningful identities, and then discusses their applications briefly. The content is summarized as follows:1. By considering the recurrence relation satisfied by infinite product and applying functional equation method, the author mainly recovers a symmetric difference formula on quintuple products, and derives a double series representation for ( q; q)1∞0. Then using modulo idea, the author gives a new proof of the Ramanujan congruence on partition function modulo 11.2. Applying mathematical induction, the author recovers a q-binomal identity with a variable x first, and then applies higher derivation to it. Basing on a derived recursive relation and using mechanized method, the author gets a generalization of Dilcher's formula, from which one can have several interesting combinatorial identities.