The Identities Of Appell-Lerch Sums And Partition Statistics

The theory of integer partitions is a significant research area in combinatorics and a branch of number theory that is closely related to other mathematical fields,including algebraic representation theory,mathematical physics,and probability theory.Many well-known mathematicians,such as Euler,Legendre,Ramanujan,Hardy,and Andrews,have made outstanding contributions to this field.The aim of this thesis is to investigate the properties of Appell-Lerch sums and partition statistics,which are currently among the most actively researched topics in the field of integer partitions.Our main research results are stated as follows:Appell-Lerch sums were first introduced by mathematicians Appell and Lerch,and played important roles in the theory of integer partitions.Zagier,a member of the American Academy of Sciences,discovered that a function can be classified as a mock theta function if it can be expressed as a linear combination of an Appell-Lerch function and a theta function with a weight of 1/2.In recent years,the congruence properties of Appell-Lerch sums have received widespread attention.In 2012,Chan posed six conjectures on congruences for Appell-Lerch sums and five of them were proved by mathematicians.In chapter 2,we confirm the remaining conjectural congruence of Chan by utilizing an identity proposed by Hickerson and Mortenson and the theory of modular forms.In order to study combinatorial interpretation of Ramanujan’s congruences for partition func-tions,Dyson,Andrews,and Garvan defined two types of classical partition statistics:rank and crank.Let N（a,m;n）and C（a,m;n）denote the number of partitions of n with rank congruent to a modulo m and the number of partitions of n with crank congruent to a modulo m,respectively.Scholars have established some equations and inequality for N（a,m;n）and C（a,m;n）,when m≤11.Recently,Aygin and Chan proved some inequality between N（a,12;n）and C（a,12;n）and proposed some conjectures.In chapter 3,applying some properties of Appell-Lerch sums and a universal mock theta function g（x,q）,we establish the generating functions for N（a,12;n）and C（a,12;n）with 0≤a≤11.With those generating functions,we obtain some new equalities and inequalities involving N（a,12;n）and C（a,12;n）and confirm several conjectures due to Aygin and Chan.With further research,statistics of some restricted integer partitions have also attracted schol-ars’attention.In 2002,Berkovich and Garvan defined the M₂-rank of partitions without repeated odd parts when they gave a new combinatorial proof of a famous identity due to Gauss.Let N₂（a,M;n）denote the number of partitions of n without repeated odd parts whose M₂-rank is congruent to a modulo M.Lovejoy,Osburn and Mao have found some formulas for M₂-rank differences modulo 3,5,6 and 10.Recently,Xia and Zhao established the generating functions for N₂（a,8;n）with 0≤a≤7.Motivated by their works,in chapter 4,we establish the generating functions for N₂（a,12;n）with 0≤a≤11 by using some identities on Appell-Lerch sums and theta function.Based on these generating functions,we prove some inequalities on N₂（a,12;n）by utilizing asymptotic formulas of eta quotients and q-series techniques.A partition is a cubic partition if its even parts come in two colours.The definition of this partition comes from Ramanujan’s"Cubic"continued fraction.Two partition statistics for cubic partitions,namely rank and crank,which provide combinatorial interpretations of congruences for cubic partitions,were defined by Kim and Reti,respectively.Let M′（r,m,n）and N′（r,m,n）denote the number of cubic partitions of n whose crank is congruent to r modulo m and the number of cubic partitions of n whose rank is congruent to r modulo m,respectively.Motivated by the works on inequalities of rank and crank for certain partitions,in chapter 5,we establish the generating functions for M′（r,m,n）and N′（r,m,n）,and then determine the signs of the differences M′（r,m,n）-M′（s,m,n）and N′（r,m,n）-N′（s,m,n）with m∈{2,3,4,6}and0≤r<s≤m-1 by utilizing asymptotic formulas of eta quotients due to Chern and q-series techniques.Finally,we conclude the paper and look forward to the follow-up research work.