Evaluation of the convolution sums ∑_{ak + bl + cm = n}σ(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) for all positive integers a,b,c,n with lcm(a,b,c)≤6. The evaluation of this convolution sum in the case (a,b,c)=(1,1,1) is due to Lahiri [17] and in the cases (a,b,c)=(1,1,2),(1,2,2) and (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=012x_j² and ∑j=16(x_2j−1²+x_2j−1x_2j+x_2j²) are reproved using new identities proved in this paper.