Congruence Properties For Two Kinds Of Partition Functions

In this paper,we mainly research two kinds of partition functions,the congruences of broken 3-diamond and Appell-Lerch sums.The main purpose of this paper is to establish more congruences of these partition functions and enrich the theory of partition and lay the foundation for future research.Broken k-diamond partition functions were introduced by Professor Geogre Andrews and Peter Paule when they investigated the MacMahon's partition theory.In Chapter 2,we establish new infinite families of congruences and non-standard congruences modulo 7 for broken 3-diamond partitions by employing some identities of Newman and the(p;k)-parametrization of theta functions due to Alaca,Alaca and Williams.Appell-Lerch sums were introduced by Appell and Lerch.Recently,a series of identities and congruences for Appell-Lerch sums were discovered by Chan.Furthermore,Chan posed several conjectures on Appell-Lerch sums at the end of his paper.In Chapter 3,we prove several congruences for Appell-Lerch sums which generalize some conjectures given by Chan by utilizing Ramanujan's theta function identities.