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 \(|q|<1\) $|q|<1$ , \(\lim _{q\rightarrow 1}(1-q)\sum _{n\ge 1} f(n) q^n \) ${lim}_{q\to 1}(1-q){\sum }_{n\ge 1}f(n)q^n$ is equal to the average value \(\lim _{N\rightarrow \infty }\) ${lim}_{N\to \infty }$ \(\frac{1}{N}\sum _{k=1}^{N}f(k)\) $\frac{1}{N}{\sum }_{k=1}^{N}f(k)$ of the sequence \(\{f(n)\}\) $\{f(n)\}$ as \(n\rightarrow \infty \) $n\to \infty$ , if the average value exists.