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 \(|q|<1\), \(\lim _{q\rightarrow 1}(1-q)\sum _{n\ge 1} f(n) q^n \) is equal to the average value \(\lim _{N\rightarrow \infty }\) \(\frac{1}{N}\sum _{k=1}^{N}f(k)\) of the sequence \(\{f(n)\}\) as \(n\rightarrow \infty \), if the average value exists.
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Notes
We do not prove general q-summability theorems here. The property must be checked for a given f(n); general proofs of q-summability would be useful. We note here, as remarks, examples from previous works [7, 8, 10] proved by less general methods, as demonstrations that our general limit theorems are not vacuous.
References
Abel, N.: Untersuchungen uber die Reihe. Theorem IV. J. Math. 1, 311–339 (1826)
Alladi, K.: Duality between prime factors and an application to the prime number theorem for arithmetic progressions. J. Number Theory 9, 436–451 (1977)
Andrews, G.E.: The Theory of Partitions, vol. 2. Cambridge University Press, Cambridge (1998)
Frobenius, G.: Über die Leibnitzsche Reihe. Journal für die Reine und Angewandte Mathematik (Crelle’s Journal) 89, 262–264 (1880)
Hardy, G.H.: Divergent Series, vol. 334. American Mathematical Soc., New York (2000)
Korevaar, J.: A century of complex Tauberian theory. Bull. Am. Math. Soc. 39(4), 475–531 (2002)
Ono, K., Schneider, R., Wagner, I.: Partition-theoretic formulas for arithmetic densities. In: Analytic Number Theory, Modular Forms and \(q\)-Hypergeometric Series, Springer Proceedings in Mathematics and Statistics 221, 611–624 (2017)
Ono, K., Schneider, R., Wagner, I.: Partition-theoretic formulas for arithmetic densities. II. Hardy-Ramanujan J. 43, 8 (2021)
Schneider, R.: Arithmetic of partitions and the \(q\)-bracket operator. Proc. Amer. Math. Soc. 145(5), 1953–1968 (2017)
Schneider, R.: Eulerian Series, Zeta Functions and the Arithmetic of Partitions, Ph.D. thesis, Emory University, (2018)
Schneider, R., Sills, A.V.: Combinatorial formulas for arithmetic density. Integers 22, 78 (2022)
Snyder, L.E.: Continuous Stolz extensions and boundary functions. Trans. Am. Math. Soc. 119, 417–427 (1965)
Acknowledgements
The author is indebted to George Andrews and Jeffrey Lagarias for discussions about analysis that influenced this paper, and to J. Lagarias for offering useful revisions; to Matthew R. Just, Ken Ono, Paul Pollack, A. V. Sills and Ian Wagner for conversations that advanced my work; and to the anonymous referee for suggestions that strengthened the final draft.
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Schneider, R. Partition-theoretic Frobenius-type limit formulas. Ramanujan J (2024). https://doi.org/10.1007/s11139-024-00835-4
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DOI: https://doi.org/10.1007/s11139-024-00835-4