| Overpartition is defined as the part of the ordinary integer partition that can be overlined or not when it first appears.It is a kind of restricted partition which plays important role in integer partition.Overpartition has a wide range of applications in combinatorics,number theory and mathematical physics.Historically,overpartition was given different names by mathematicians from different branches.Until 2002,Corteel and Lovejoy defined this kind of partition as overpartition and conducted a systematic study.The aim of this thesis is to investigate congruence properties for overpartition and its statistics:rank and crank.Congruence properties for overpartition were first studied by Hirschhorn and Sellers.Since then,many congruences modulo powers of 2,3,5 have been discovered.Recently,Xia proved several infinite families of congruences modulo 9 and 27 and posed a conjecture on congruence modulo 243 for overpartition.In chapter 2,we use the(p,k)parameter of theta functions to prove the conjecture proposed by Xia.In order to present combination interpretation of congruences modulo 5,7,11 for ordinary partition function given by Ramanujan,Dyson,Andrews and Garvan gave the definitions of rank and crank for ordinary partition.Inspired by their work,Bringmann,Lovejoy and Osburn gave the definitions of rank and crank of overpartition.Let N(s,l;n),M(s,l;n),M2(s,l;n)denote the number of overpartition of n with the rank,the first residual crank,the second residual crank congruent to s modulo l.Using the properties of Appell-Lerch sums,we establish the generating functions of N(s,l;n),M(s,l;n)and M2(s,l;n)where l ∈ {4,8} and 0<s<l.Based on these generating functions,we prove some equalities and inequalities of N(s,l;n),M(s,l;n)and M2(s,l;n)with l ∈ ｛4,8}. |
