Detailed asymptotic expansions for partitions into powers

TL;DR

This paper derives detailed asymptotic expansions for the number of partitions of large integers into powers, proving convexity, log-concavity, and confirming conjectures related to these partitions.

Contribution

It provides simplified proofs of Wright's asymptotic results, exact formulas for expansion coefficients, and confirms conjectures on convexity and log-concavity.

Findings

Convexity and log-concavity of partition counts for large n

Exact formulas for asymptotic expansion coefficients

Proof of Ulas's stronger conjectures

Abstract

Here we examine the number of ways to partition an integer $n$ into $k$th powers when $n$ 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 $n$, and the stronger conjectures of Ulas are proved. The asymptotics of Wright's generalized Bessel functions are also treated.