  If $f(z)$ is a modular form of weight $k$, then the differential operator
$\vartheta_k$ defined by $\vartheta_k(f) = \frac{1}{2\pi i} \frac{d}{dz}f(z) -
\frac{k}{12} E_2(z) f(z)$ (known as the Ramanujan-Serre derivative map) is a
modular form of weight $k+2$. In this paper, we obtain a simple extension of
this map and use it to get a general method to derive certain convolution sums
of the divisor functions (using the theory of modular forms). Explicit
expressions are given for four types of convolution sums and we provide many
examples for all these types of sums.

Number Theory