Exact Formulae for the Fractional Partition Functions

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 Turán inequalities) that $p_{\alpha }(n)$ satisfies.

Fixed typos and made minor stylistic changes