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arXiv · 2011.08386

Partition-theoretic Frobenius-type limit formulas

Robert Schneider
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Abstract

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

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Robert Schneider. 2020-11-17. Partition-theoretic Frobenius-type limit formulas. https://doi.org/10.1007/s11139-024-00835-4

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