Evaluation of the convolution sums ∑ak + bl + cm = nσ(k)σ(l)σ(m) with lcm(a,b,c)≤6

The generating functions of divisor functions are quasimodular forms and their products belong to a space of quasimodular forms of higher weight. In this work, we evaluate the convolution sums ∑ak+bl+cm=nσ(k)σ(l)σ(m)∑ak+bl+cm=nσ(k)σ(l)σ(m) for all positive integers a,b,c,na,b,c,n with lcm(a,b,c)≤6lcm(a,b,c)≤6. The evaluation of this convolution sum in the case (a,b,c)=(1,1,1)(a,b,c)=(1,1,1) is due to Lahiri [17] and in the cases (a,b,c)=(1,1,2),(1,2,2)(a,b,c)=(1,1,2),(1,2,2) and (1,2,4)(1,2,4) to Alaca, Uygul and Williams [7]. As an application, the known formulas for the number of representations of a positive integer n by each of the quadratic forms∑j=012xj2 and ∑j=16(x2j−12+x2j−1x2j+x2j2) are reproved using new identities proved in this paper.