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arXiv · 1907.03026

Exact Formulae for the Fractional Partition Functions

Jonas Iskander·Vanshika Jain·Victoria Talvola
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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 by $\sum_{n = 0}^\infty p_{\alpha}(n)x^n := \prod_{k=1}^\infty (1-x^k)^{-\alpha}$. In this paper we use the Rademacher circle method to find an exact formula for $p_\alpha(n)$ and study its implications, including log-concavity and the higher-order generalizations (i.e., the Tur\'an inequalities) that $p_\alpha(n)$ satisfies.

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Jonas Iskander, Vanshika Jain, Victoria Talvola. 2019-07-05. Exact Formulae for the Fractional Partition Functions. https://arxiv.org/abs/1907.03026

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