On The Number Of Representations Of N As A Linear Combination Of Four Triangular Numbers

Let Z and N be the set of integers and the set of positive integers,respectively.For a,b,c,d,n ?N let(a,b,c,d;n)be the number of representations of n by ax(x-1)/2 + by(y-1)/2 + cz(z-1)/2 + dw(w-1)/2(x,y,z,w ? Z).In this thesis,by using elementary methods and identities for Ramanujan's theta functions,we obtain explicit formulas for,(a,b,c,d;n)in the 21 cases(a,b,c,d)=(1,2,2,4),(1,2,4,4),(1,1,4,4),(1,4,4,4),(1,3,9,9),(1,1,3,9),(1,3,3,9),(1,1,9,9),(1,9,9,9),(1,1,1,9),(1,1,2,8),(1,1,2,16),(1,2,3,6),(1,3,4,12),(1,1,3,4),(1,1,5,5),(1,5,5,5),(1,3,3,12),(1,1,1,12),(1,1,3,12),(1,3,3,4).In addition,we prove that t(a,b,c,d;n)= 2/3N(a,b,c,d;8n + a + b + c + d)-2N(a,b,c,d;2n+(a+b+c+d)/4)in the cases(a,b,c,d)=(a,a,2a,8m),(a,3a,8k+2,8m+6),(a,3a,8m + 4,8m+ 4)(n?m+a-1/2(mod 2))and(a,3a,16k + 4,16m+4)(n?a-1/2(mod 2)),where a is a positive odd integer,m ? N and k ? {0,1,2,...}.