On Combinatorial Congruences And Supercongruences For Truncated Hypergeometric Series

Combinatorial congruences belong to the field of Combinatorial Number Theory,which were investigated by lots of famous mathematicians.In this thesis we mainly discuss combinatorial congruences and supercongruences for truncated hypergeometric series.Combinatorial congruences are related to many fields of mathematics,such as p-adic analysis and hypergeometric series,and they even have a deep connection with algebraic topology.In this thesis,by using tools of Zeilberger algorithm,combinatorial identities,Bernoulli numbers,Euler numbers and hypergeometric series and so on,we mainly prove the following few congruences modulo powers of a prime p conjectured by Z.-W.Sun:(?)If p=3(mod 4),then (?)(?)If p?1(mod 12),then (?) where x3(k)denotes the Legendre symbol(k/3).(?)If p?7(mod 8),then (?) where the Pell numbers Pn are given by PO=0,P1=1,Pn=2Pn-1+Pn-2(n=2,3,…).(iv)If p ? 11(mod 12),then where the numbers Rn are given by R0=2,R1=4,Rn=4Rn-1-Rn-2(n = 2,3,…).And for any odd prime p,we have(?)Pk/8k?1+2(-1)(p-1)/2p2Ep-3(mod p3),(?)Pk/16k?(-1)(p-1)/2-p2Ep-3(mod p3),where Pn=(?)is the n-th Catalan-Larcombe-French number and Ep-3 is the(p-3)-th Euler number.We also prove a conjecture of Deines,Fuselier,Long,Swisher and Tu[12,Conjec-ture 18]:For any prime p?1(mod 4),we have where(?)k denotes ?(?+1)…(?+k-1)for k? 1 and(?)0=1.We also prove some other congruences conjectured by Z.-W.Sun and some con-gruences conjectured by V.J.W.Guo and J.-C.Liu.