Identities involving sum of divisors, integer partitions and compositions

Abstract

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
$$\frac{(q^{a_1};q^{b_1}{)}_{\mathrm{\infty }}(q^{a_2};q^{b_2}{)}_{\mathrm{\infty }}\cdots (q^{a_k};q^{b_k}{)}_{\mathrm{\infty }}}{(q^{c_1};q^{d_1}{)}_{\mathrm{\infty }}(q^{c_2};q^{d_2}{)}_{\mathrm{\infty }}\cdots (q^{c_r};q^{d_r}{)}_{\mathrm{\infty }}},$$
where $a_1,\dots ,a_k,b_1,\dots ,b_k,c_1,\dots ,c_r,d_1,\dots ,d_r$ are positive integers, and $|q|<1$.