  One of the main goals in this paper is to establish convolution sums of
functions for the divisor sums
$\widetilde{\sigma}_s(n)=\sum_{d|n}(-1)^{d-1}d^s$ and
$\widehat{\sigma}_s(n)=\sum_{d|n}(-1)^{\frac{n}{d}-1}d^s$, for certain $s$,
which were first defined by Glaisher. We first introduce three functions
$\mathcal{P}(q)$, $\mathcal{E}(q)$, and $\mathcal{Q}(q)$ related to
$\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$,
respectively, and then we evaluate them in terms of two parameters $x$ and $z$
in Ramanujan's theory of elliptic functions. Using these formulas, we derive
some identities from which we can deduce convolution sum identities. We discuss
some formulae for determining $r_s(n)$ and $\delta_s(n)$, $s=4,$ $8$, in terms
of $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and
$\widetilde{\sigma}_3(n)$, where $r_s(n)$ denotes the number of representations
of $n$ as a sum of $s$ squares and $\delta_s(n)$ denotes the number of
representations of $n$ as a sum of $s$ triangular numbers. Finally, we find
some partition congruences by using the notion of colored partitions.
