Extensions of MacMahon's sums of divisors

In 1920, P. A. MacMahon generalized the (classical) notion of divisor sums by relating it to the theory of partitions of integers. In this paper, we extend the idea of MacMahon. In doing so we reveal a wealth of divisibility theorems and unexpected combinatorial identities. Our initial approach is quite different from MacMahon and involves rational function approximation to MacMahon-type generating functions. One such example involves multiple $q$-harmonic sums $$\sum_{k=1}^n\frac{(-1)^{k-1}\genfrac{[}{]}{0pt}{}{n}{k}_{q}(1+q^k)q^{\binom{k}{2}+tk}}{[k]_q^{2t} \genfrac{[}{]}{0pt}{}{n+k}{k}_{q}}=\sum_{1\leq k_1\leq\cdots\leq k_{2t}\leq n}\frac{q^{n+k_1+k_3\cdots+k_{2t-1}}+q^{k_2+k_4+\cdots+k_{2t}}}{[n+k_1]_q[k_2]_q\cdots[k_{2t}]_q}.$$