Partition-theoretic Frobenius-type limit formulas

TL;DR

This paper develops partition-theoretic methods to establish q-series analogues of Frobenius' limit formula, extending classical results on power series convergence to a broader, partition-based framework.

Contribution

It introduces new partition-generating function techniques to derive Frobenius-type limit formulas for q-series, generalizing Abel's convergence theorem.

Findings

Derived q-series analogues of Frobenius' limit formula.

Extended Abel's convergence theorem to partition-generating functions.

Established new connections between partition theory and complex analysis.

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.