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.
Categories: math.NT math.CO