On Evaluation of Convolution Sums Involving Divisor Functions and Partition Functions

In this article, we mainly focus on answering the open problem given by Huard, Ou, Spearman, and Williams in (Number Theory for the Millennium, II (Urbana, IL, 2000), pp. 229-274, 2002) which many have solved partially. We solve this problem using elementary method involving q-series identities and Ramanujan's P-Q relations. On the course of solving this open problem, we also find new interesting convolution sums of higher order. As an application we determine exclusive formulae for the number of representation of a positive integer n by the quadratic formsa(x(1)(2)+x1x(2)+x(2)(2)+x(3)(2)+x3x(4)+x(4)(2))+b(x(5)(2)+x5x(6)+x(2)5+x(7)(2)+x7x(8)+x(8)(2))+c(x(9)(2)+x(9)x10+x(10)(2)+x(2)11+x(11)x(12)+x(12)(2))+d(x(2)13+x(13)x(14)+x(14)(2)+x(15)(2)+x(15)x(16)+x(16)(2))for (a,b,c,d)=(1,2,2,0)=(1,2,2,0), (2, 1, 1, 0), (1, 1, 1, 1), and (1, 1, 2, 2).