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 \( n > 0 \), \[ ∑ₗ₌₁^n 1/l! ∑w_1+w_2+⋯+w_l ∈ C(n) σ_1(w_1) σ_1(w_2) ⋯ σ_1(w_l)/w_1 w_2 ⋯ w_l = p_1(n), \] where \( σ_1(n) \) 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 \[ {(qa_1;qb_1)_∞ (qa_2;qb_2)_∞ ⋯ (qa_k;qb_k)_∞} {(qc_1;qd_1)_∞ (qc_2;qd_2)_∞ ⋯ (qc_r;qd_r)_∞}, \] where \( a_1, …, a_k, b_1, …, b_k, c_1, …, c_r, d_1, …, d_r \) are positive integers, and \( |q| < 1 \).