Congruences On Combinatorial Numbers And Hypergeometric Series

In the past few years,many congruences on sums of combinatorial numbers have been widely studied.Various supercongruences have been also conjectured by many authors,including Beukers,van Hamme,Rodriguez-Villegas,Zudilin,Z.-W.Sun,Z.-H.Sun and Guo.In this thesis,we will prove some congruences involving Delannoy number-s,Schroder numbers,Catalan numbers,Schroder-like numbers,central binomial coefficients and q-binomial coefficients,which were conjectured by Apagodu,Zeil-berger,Z.-W.Sun and Guo.For example,we prove that for any prime p ? 5 and positive integer n,which extends a result of Sun and Tauraso,and was originally conjectured by Apagodu and Zeilberger.To establish the relationship between the number of points over Fp on hyper-geometric Calabi-Yau manifolds and truncated hypergeometric series,Rodriguez-Villegas conjectured 22 interesting supercongruences for truncated hypergeomet-ric series.Some of these conjectures were gradually proved by several authors,including van Hamme,Ishikawa,Ahlgren,Mortenson,Fuselier,McCarthy,Kil-bourn and Z.-W.Sun.In this thesis,we will also generalize the Rodriguez-Villegas supercongruences for truncated hypergeometric series ₂F₁ and ₃F₂.