Detailed asymptotic expansions for partitions into powers

Abstract

In this paper, we examine the number of ways to partition an integer [Formula: see text] into [Formula: see text]th powers when [Formula: see text] is large. Simplified proofs of some asymptotic results of Wright are given using the saddle-point method, including exact formulas for the expansion coefficients. The convexity and log-concavity of these partitions is shown for large [Formula: see text], and the stronger conjectures of Ulas are proved. The asymptotics of Wright’s generalized Bessel functions are also treated.