Partition-theoretic Frobenius-type limit formulas

Using partition generating function techniques, we prove $q$-series analogues of a formula of Frobenius generalizing Abel’s convergence theorem for complex power series. Frobenius’ result states that for $\mathrm{\mid }q\mathrm{\mid }<1$, $\lim_{q\to 1}(1-q)\sum_{n\geq 1} f(n) q^n $isequaltotheaveragevalue$\lim_{N\to \infty}$ $\frac{1}{N}{\sum }_{k=1}^{N}f(k)$ of the sequence $\{f(n)\}$ as $n\to \infty$, if the average value exists.