The M2-rank Of Partitions Without Repeated Odd Parts And The Turncated Sums Of Theta Functions

The theory of integer partition lies at the forefront of research in combinatorics and number theory and attracts the attention of many famous mathematicians such as George Andrews,a members of the United States National Academy of Sciences.The aim of this thesis is to investigate the M₂-rank of partitions without repeated odd parts and truncated sums for classical theta functions.Our research can be stated as follows.In 2002,Berkovich and Garvan introduced the M₂-rank of partitions without repeated odd parts.Let N₂（a,M,n）denote the number of partitions of n without repeated odd parts whose M₂-rank is congruent to a mod M.Lovejoy,Osburn and Mao found a number of nice results for M₂-rank differences modulo 3,5,6 and 10.In the chapter 2,by using some properties for Appell-Lerch sums,we establish the generating functions for N₂（a,8,n）with 0≤a≤7.With these generating functions,we obtain some equalities and inequalities on M₂-rank modulo 8 of partitions without repeated odd parts.We also relate some differences of the M₂-rank to eighth-order mock theta functions.In 2012,Andrews and Merca derived a truncated version of Euler’s pentagonal number theorem.Their work inspired several mathematicians to work on truncated theta series including Guo and Zeng,who examined two other classical theta series identities of Gauss.In chapter 3,revisiting these three theta series identities of Euler and Gauss,we obtain new truncated theorems.As corollaries of our results,we obtain infinite families of linear inequalities involving the partition function,the overpartition function and the pod function.These inequalities yield the positive result of Andrews and Merca on the partition function as well as a conjecture on the overpartition function,which was posed by Andrews–Merca and Guo–Zeng.We will also provide a unified combinatorial treatment for our results.In chapter 4,using the Rogers-Fine identity,we present a partition theoretic interpretation of truncated sums on the partition function which solve a problem given by Merca.Furthermore,we prove an inequality on ordinary partition which implies the positive result on truncated sums of partition function due to Andrews and Merca.To conclude this thesis,we present a summary and pose several problems which will be investigated in future.