The second shifted difference of partitions and its applications

TL;DR

This paper establishes effective bounds on ratios of partition functions and the second shifted difference of partitions, leading to new inequalities and properties relevant to partition theory.

Contribution

It introduces new bounds and estimates for partition ratios and differences, including applications to shifted convexity and partition functions.

Findings

Proved an effective bound on $p(n-j)/p(n)$ ratios.

Derived an estimate for the second shifted difference $f(j,n)$ of partitions.

Established a shifted convexity property of the partition function $p(n)$.

Abstract

A number of recent papers have estimated ratios of the partition function $p(n-j)/p(n)$, which appears in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study second shifted difference of partitions, $f(j,n):= p(n) -2p(n-j) +p(n-2j)$, and give another easy-to-use estimate of $f(j,n)$. As applications of these, we prove a shifted convexity property of $p(n)$, as well as giving new estimates of the $k$-rank partition function $N_k(m,n)$ and non-$k$-ary partitions along with their differences.