Evaluation of the convolution sums Σa1m1+a2m2+a3m3+a4m4=n σ(m1)σ(m2)σ(m3)σ(m4) with 1cm(a1, a2, a3, a4) ≤ 4

The generating functions of divisor functions are quasimodular forms of weight 2 and the product of them is a quasimodular form of higher weight. In this work, we evaluate the convolution sums Sigma(a1m1+a2m2+a3m3+a4m4=n) sigma(m(1))sigma(m(2))sigma(m(3))sigma(m(4)) for the positive integers a(1), a(2), a(3), a(4), and n with lcm (a(1), a(2), a(3), a(4)) <= 4. We reprove the known formulas for the number of representations of a positive integer n by each of the quadratic forms Sigma(16)(j=0) x(j)(2) and Sigma(8)(j=1) (x(2j-1)(2) + x(2j-1)x(2j) + x(2j)(2) as an application of the new identities proved in this paper.