Some Results On Partition Functions And The Expansions Of Infinite Products

In this thesis,we aim to prove four new congruences modulo 5 for pw(n),two infinite families of congruences modulo 5 and 27 for t(n).Besides,we investigate the properties of coefficients in the power series expansions of three couples of infinite products.The thesis is organized as follows.In chapter 1,we introduce the background of the theory of partition,the research status at home and abroad related to the content of this thesis,and the main work of this thesis.In chapter 2,we recall the concept of several partition functions and theta function-s,and introduce some identities involving theta functions and some necessary lemmas.In chapter 3,based on an identity related to a mock theta function of order 3 established by Watson,applying some relevant q-series identities including Jacobi's identity,we prove four new congruences modulo 5 for pw(n).In chapter 4,from the generating function of t(n)due to Bringmann et al,applying Jacobi's identity,Euler's pentagonal number theorem and related lemmas,we derive two infinite families of congruences modulo 5 and 27 for t(n).In chapter 5,we consider three couples of infinite products,and establish the 5-dissections of these products,from which we get the properties of coefficients in power series expansions of these infinite products,including the vanishing coefficients of arithmetic progressions and the periodicity of the signs of the coefficients.In chapter 6,we review and summarize some main work of this thesis.