THE SECOND SHIFTED DIFFERENCE OF PARTITIONS AND ITS APPLICATIONS

2022/08/25 by KEVIN GOMEZ, Kevin Gomez, JOSHUA MALES +3 · 3 citations
Mathematics · #Advanced Mathematical Identities #Advanced Combinatorial Mathematics #Mathematical Inequalities and Applications

paper · doi:10.1017/s0004972722000764

Abstract

Abstract A number of recent papers have estimated ratios of the partition function p(n-j)/p(n) , which appear in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study the 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.

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