Exact formulae for the fractional partition functions
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Author(s)
Iskander, Jonas
Jain, Vanshika
Talvola, Victoria
Date Issued
April 21, 2020
Publisher
Springer International Publishing
Citation
Research in Number Theory. 2020 Apr 21;6(2):20
Version
Author's final manuscript
Abstract
Abstract
The partition function p(n) has been a testing ground for applications of analytic number theory to combinatorics. In particular, Hardy and Ramanujan invented the “circle method” to estimate the size of p(n), which was later perfected by Rademacher who obtained an exact formula. Recently, Chan and Wang considered the fractional partition functions, defined for $$\alpha \in {\mathbb {Q}}$$α∈Q by $$\sum _{n = 0}^\infty p_{\alpha }(n)x^n := \prod _{k=1}^\infty (1-x^k)^{-\alpha }$$∑n=0∞pα(n)xn:=∏k=1∞(1-xk)-α. In this paper we use the Rademacher circle method to find an exact formula for $$p_\alpha (n)$$pα(n) and study its implications, including log-concavity and the higher-order generalizations (i.e., the Turán inequalities) that $$p_\alpha (n)$$pα(n) satisfies.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s40993-020-00195-0
