Hypergeometric Series Method For Riemann-Zeta Function And Combinatorial Identities

By means of combination of classical analysis, hypergeometric series and formal power series method, this dissertation investigates the problems on combinatorial computations of closed formulae of Riemann-Zeta function, infinite series identities as well as Pascal matrices, etc. The contents is as follows:1. Based on two hypergeometric expansion formulae of trigonometric functions, we combine their derivatives with the symmetric functions and establish numerous infinite series identities involving the harmonic numbers. For the presence of free parameters, the hypergeometric formulae that we apply possess strong flexibility. This allows us, successfully and systematically, to treat a class of infinite series identities. In particular, by the Gauss hypergeometric series we obtain a series of closed forms of infinite series with central binomial coefficients in denominator, which extends essentially the related results due to Eisner (2005).2. In 1997, Chu discovered the hypergeometric series method for Riemann-Zeta function. As continuation and extension of this approach, we apply the summation theorems due to Gauss, Kummer and Dixon to derive a large series of summation formulae related to Riemann-Zeta function ζ(5) and ζ(6). The simple form and the variety of the results obtained show again that the hypergeometric series method is indeed powerful for studying Riemann-Zeta function.3. Under the appropriate parameter replacements in the hypergeometric summation formulae named after Gauss, Whipple and Watson, the hypergeometric method enables us to deal with another class of infinite series summation formulae. The summands of this class of infinite series are simple linear functions of central binomial coefficients and higher harmonic numbers. Their characteristic lies in the close relation to Riemann-Zeta function. This differs substantially from the infinite series mentioned before.4. The q-analogues of the Pascal matrix and the symmetric Pascal matrix are studied. It is shown that the q-Pascal matrix can be factorizcd by special matrices and the symmetric q-Pascal matrix has the Cholesky factorization. We get a close relation between q-Pascal matrix and symmetric q-Pascal matrix. The determinants of these matrices arc also computed. Furthermore the expressions for powers and exponential functions of these matrices are obtained...