K. Uchimura [Discrete Appl. Math. 18, 73-81 (1987; Zbl 0629.10005)] studied the \(q\)-series \[ U_m(q)= \sum^\infty_{n= 1} n^m q^n \prod^\infty_{j= n+ 1} (1- q^j). \] The author proves that the generating functions of the divisor functions \(\sigma_k(n)= \sum_{d|n} d^k\) are expressed as sums of products of \(U_m(q)\) \((m= 1,\dots, k+ 1)\) and vice versa. It is also shown that the \(U_m(q)\) are closely related to certain recurrences of polynomials. Other related \(q\)-series identities are derived, including a class of binomial sums connected with the harmonic numbers.