Congruences Involving Combinatorial Sequences And Truncated Hypergeometric Series

Congruences related to combinatorial sequences have been widely studied during the past few centuries.In this thesis,we prove some congruences involving combinatorial sequences such as binomial coefficients,Apéry numbers and Franel numbers.For example,using congruence properties of the harmonic numbers,we prove that for any prime p? 7,(?)2k+1)Ak?p+7/2p2Hp-1(mod p6),and for any prime p?5,(?)(2k+1)3Ak?p3+4p4Hp-1+6/5p8Bp-5(mod p9),where(?)is the kth Apery number,Hp-1=?k=1p-1/k is the(p-1)th harmonic number and Bp-5 is the(p-5)th Bernoulli number.These confirm two conjectures of Z.-W.Sun.For n ? N={0,1,2...},we define the truncated hypergeometric series#12 where#12 Clearly,the truncated hypergeometric series is a finite analogue of the classical hypergeometric series.Congruences involving truncated hypergeometric congruences are closely related to Calabi-Yau manifolds,elliptic curves,modular forms and p-adic Gamma functions.In this thesis,by using some known identities for hypergeometric series,the Wilf-Zeilberger method and the symbolic summation method,we obtain some new hypergeometric congruences.Let n ? 3 be an integer and p be a prime with p?1(mod n).With the help of the celebrated Karlsson-Minton summation formula,we prove that#12 and#12 where ?p is the p-adic Gamma function.These two congruences confirm two conjectures proposed by A.Deines,J.G.Fuselier,L.Long,H.Swisher and F.-T.Tu.In this thesis,we also discover some curious hypergeometric identities.For example,for any n ? N and ? ? {0,1},we find that#12 Using this identity,we confirm the following conjecture of Z.-W.Sun:(?)(mod p2)for any prime p>3,which is closely related to Z.-W.Sun's conjectural series identity#12 where(·/3)is the Legendre symbol.Moreover,we also prove some other congruences conjectured by Z.-W.Sun and V.J.W.Guo.