Congruences For Partition Functions And Hecke-Rogers Type Identities

The main purpose of this thesis is to study the congruence properties of some partition functions and Hecke-Rogers type series identities,which are two important parts in the field of basic hypergeometric series.For the first part,we establish some interesting Ramanujan-type congruences for functions related to the 4-colored generalized Frobenius partitions,2-color partition triples,and a second order mock theta function.The main tools used in our proofs include Jacobi triple product identity,the quintuple product identity,Eisenstein series and the powerful algorithm due to Radu and Sellers which involves modular forms.We also refine the Watson-Atkin style method to give a completely elementary proof for congruences modulo powers of a prime.The second part is devoted to the study of Hecke-Rogers type series identities with the help of a q-series expansion formula due to Liu.For one thing,We establish certain q-transformation formulas and discuss their applications to Rogers-Ramanujan type identities and Hecke-Rogers type identities.For another,inspired by the recent work of Andrews-Merca and Guo-Zeng on the truncated version of theta functions,we generalize this topic to Hecke-Rogers type double series,which are associated with some interesting partition functions.In addition,we reprove an ex-conjecture of Andrews-Merca and Guo-Zeng on the truncated Jacobi triple product series by providing an explicit series form.A companion theorem to this ex-conjecture is also presented.