  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.
