Abstract
AbstractA 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.
Publisher
Cambridge University Press (CUP)