Partition-theoretic Frobenius-type limit formulas
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866913312298500096 |
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| author | Schneider, Robert |
| author_facet | Schneider, Robert |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_08386 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Partition-theoretic Frobenius-type limit formulas Schneider, Robert Number Theory Combinatorics 05A17 (Primary), 40E05, 11P82 (Secondary) 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. |
| title | Partition-theoretic Frobenius-type limit formulas |
| topic | Number Theory Combinatorics 05A17 (Primary), 40E05, 11P82 (Secondary) |
| url | https://arxiv.org/abs/2011.08386 |