Identities involving sum of divisors, integer partitions and compositions

In this paper we show some identities come from the \( q \)-identities of Euler, Jacobi, Gauss, and Rogers-Ramanujan. Some of these identities relate the function sum of divisors of a positive integer \( n \) and the number of integer partitions of \( n \). One of the most intriguing results found here is given by the next equation, for \( n > 0 \),
\[
\sum_{l=1}^n \frac{1}{l!} \sum_{w_1+w_2+\cdots+w_l \in C(n)} \frac{\sigma_1(w_1) \sigma_1(w_2) \cdots \sigma_1(w_l)}{w_1 w_2 \cdots w_l} = p_1(n),
\]
where \( \sigma_1(n) \) is the sum of all positive divisors of \( n \), \( p_1(n) \) is the number of integer partitions of \( n \), and \( C(n) \) is the set of integer compositions of \( n \). In the last section, we show seven applications, one of them is a series expansion for
\[
\frac{(q^{a_1};q^{b_1})_\infty (q^{a_2};q^{b_2})_\infty \cdots (q^{a_k};q^{b_k})_\infty}
{(q^{c_1};q^{d_1})_\infty (q^{c_2};q^{d_2})_\infty \cdots (q^{c_r};q^{d_r})_\infty},
\]
where \( a_1, \ldots, a_k, b_1, \ldots, b_k, c_1, \ldots, c_r, d_1, \ldots, d_r \) are positive integers, and \( |q| < 1 \).