Applications Of Modular Functions In The Ramanujan-type Identities And Congruences

The Ramanujan-type identities and congruences of the partition functions are classical topics in combinatorics and number theory.There are a number of meth-ods to derive such identities and congruences,including analytical methods,basic hypergeometric methods,combinatorial arguments and algorithmic approaches.This thesis is mainly concerned with a class of partition functions a(n)defined in terms of generalized eta-quotients by algorithmic approaches,and Ramanujan's general partition functions p_r(n)by using the analytical methods.In Chapter 1,we first introduce the research background on the Ramanujan-type identities and congruences for the partition function p(n)and Ramanujan's gen-eral partition functions p_r(n),then we focus on the algorithmic approaches to such identities and congruences.In Chapter 2,we present an algorithm to find Ramanujan-type identities for a(mn+t)based on the theory of modular functions with respect to?₁(N)for given integers m>0 and 0?t?m-1.As applications,we deduce a witness identity for p(11n+6)with integer coefficients,and the Ramanujan-type identities for the overpartition function (?)(n),Andrews–Paule's broken 2-diamond partition function?₂(n),Andrews'(3,1)-singular overpartition function (?)_3,1(n).Our algorithm also yields the 2-dissection formulas of Ramanujan and the 8-dissection formulas due to Hirschhorn.In Chapter 3,we develop an algorithm to the Ramanujan-type congruences for a(n)by using the theory of modular forms for?₁(N)and Sturm's theorem.As applications,some congruences for Andrews'(k,i)-singular overpartition functions are obtained.In Chapter 4,we present a unified approach to establish the Ramanujan-type identities and congruences for p_-r(n)by using the U-operator,the theory of mod-ular functions,and a result of Engstrom about the periodicity of linear recurring sequences modulo any integer,where 1?r?24 and 3|r or 8|r.As applications,in-finite families of congruences for many partition functions such as (?)(n)and l-regular partition functions b_l(n)are easily obtained.