Exact formulae for the fractional partition functions

Abstract

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 for $$\alpha \in {\mathbb {Q}}$$α∈Q by $$\sum _{n = 0}^\infty p_{\alpha }(n)x^n := \prod _{k=1}^\infty (1-x^k)^{-\alpha }$$∑n=0∞pα(n)xn:=∏k=1∞(1-xk)-α. In this paper we use the Rademacher circle method to find an exact formula for $$p_\alpha (n)$$pα(n) and study its implications, including log-concavity and the higher-order generalizations (i.e., the Turán inequalities) that $$p_\alpha (n)$$pα(n) satisfies.