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Exact Formulae for the Fractional Partition Functions

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 $\sum_{n = 0}^\infty p_α(n)x^n := \prod_{k=1}^\infty (1-x^k)^{-α}$. In this paper we use the Rademacher circle method to find an exact formula for $p_α(n)$ and study its implications, including log-concavity and the higher-order generalizations (i.e., the Turán inequalities) that $p_α(n)$ satisfies.