# Exact Formulae for the Fractional Partition Functions

## TL;DR

This paper derives exact formulas for fractional partition functions using the circle method, extending classical results and exploring properties like log-concavity and Turán inequalities.

## Contribution

It introduces a novel application of the Rademacher circle method to fractional partition functions, providing explicit formulas and analyzing their mathematical properties.

## Key findings

- Derived exact formulas for fractional partition functions.
- Established log-concavity and Turán inequalities for these functions.
- Extended classical partition theory to fractional cases.

## 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.