Research On Congruence Properties For Partition Functions Based On Modular Forms

Modular forms have played a vital role in the study of partition functions and is one of the important tool for the study of congruence properties of partition functions.The main content of this thesis is to study four types of derivative partition functions by using the theory of modular form.This thesis is organized as follows:In the first part,by using the results of two types of η-quotients under the action of specific Hecke operators,we prove the congruences of two types of powers of the Delta functions.In this way we obtain two types of congruences for rth powers of the partition,one of which is a Ramanujan-type congruence.In the second part,we study the congruence properties of the broken k-diamond partition.By establishing the congruences between the generating functions and certain Hecke eigenforms,and inspired by Ogasawara’s work,we prove four types infinite congruences families for broken k-diamond partition.Furthermore,we also study the k-dots bracelet partition,which is a generalization of the broken k-diamond partition.We obtain two congruences for the k-dots bracelet partition.In the third part,we study the congruence properties of t-tuples partition with odd parts distinct.By establishing the congruences between the generating functions and some η-quotients,and using the theta function theory,we obtain six types infinite congruences families for t-tuples partition with odd parts distinct.In the fourth part,we study the congruence properties of two classes of l-regular partitions with certain restrictions.By using the logarithmic operator and Boylan’s theorem,we prove a convolution and a congruence for l-regular partition with odd parts distinct.And we also obtain a congruence for l-regular partition with even parts distinct.At the same time,we establish a congruence relationship which connects l-regular partition with odd parts distinct and l-regular partition with even parts distinct.