The Representation Number Of A Positive Integer By Sums Of Mixed Numbers

The representation number of a positive integer by sums of mixed numbers is one of the most active filed in contemporary combinatorics and number theory,which has numerous natural connections to many branches of mathematics,that is exactly the reason why many well-known combinatorialists,including Johann Carl Gauss,are interested in it.In this thesis,we mainly investigate the representation number of a positive integer by sums of mixed numbers which have six variables and eight variables.Let N(r,s;t;n)and N(a1,a2,a3;b1,b2;c1,c2,c3,c4;n)denote the number of representations of a positive integer n by the form with six variables and eight variables respectively.Utilizing the(p,k)parametrization of theta functions and Eisenstein series due to A.Alaca,S.Alaca and K.S.Williams,explicit formulas for N(r,s;t;n)and N(a1,a2,a3;b1,b2;c1,c2,c3,c4;n)are determined for any positive integer n and special integers ?r,s,t} and {a1,a2,a3;b1,b2;c1,c2,c3,c4}.This thesis is organized as follows:In chapter 1,we introduce the research background,the research progress and the main research work of this thesis.In chapter 2,we introduce some theta functions,Eisenstein series and their(p,k)parametrization which will be employed to establish new identities involving theta functions and Eisenstein series in chapters 3 and 4.In chapter 3,a series of new theta function identities are established by using the(p,k)parametrization of theta functions.Utilizing these identities,we investigate the representation number of a positive integer by sums of mixed numbers which have six variables and establish explicit formulas for N(2,0;1;n),N(2,0;2;n),N(2,0;4;n),N(1,1;2;n),N(1,1;4;n),N(0,2;1;n),N(0,2;2;n)?N(0,2;4;n).In chapter 4,we define G(q)and H(q).By utilizing the(p,k)parametrization of Eisenstein series,we deduce several new identities involving Eisenstein series and study the representation number of n by sums of mixed numbers which has eight variables.From these identities,we characterize some explicit formulas of N(a1,a2,a3;b1,b2;c1,c2,c3,c4;n).At the end of this thesis,we will summarize our work and look forward to the future work.