Research On Congruences, Theta Functions And Pi

In 1996. L. van Hamme presented 13 congruences linking the partial sum-s of certain hypergeometric series to the values of the p-adic Gamma function and they were denoted by (A.2)-(M.2).He proved 3 congruences, namely, (C.2), (H.2) and (I.2) and gave a weaker result of another one. In the remaining 10 congruences, some of them were confirmed by different other authors. Later, Kil-bourn, McCarthy and Osburn, Mortenson, Long, Long and Ramakrishna settled (M.2), (A.2), (B.2), (J.2) and (D.2) respectively. In this thesis, we employ p-adic Gamma function and formulas on hypergeometric series to prove (E.2), (F.2) and (G.2). We also give another supercongruence which includes as a special case one of Rodriguez-Villegas’conjectures.Z.-W. Sun proposed several conjectures on congruences and divisibility prop-erties of binomial sums. In this thesis, we use the WZ method to confirm four conjectures on the divisibility of binomial sums and a supercongruence conjecture of Z.-W. Sun, set up two other new divisibility properties and give another proof of a supercongruence.In addition, we give some congruences on q-Catalan numbers and q-harmonic numbers, derive some nontrivial identities on circular summation of theta func-tions and some finite trigonometric sum identities from an equivalent form of Liu’s identity and employ certain representations for Eiscnstein series, identities for hypergeometric series and Ramanujan’s quartic theory of elliptic functions to derive several new series for 1/π2.