THE SECOND SHIFTED DIFFERENCE OF PARTITIONS AND ITS APPLICATIONS

Abstract A number of recent papers have estimated ratios 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, then study second shifted difference partitions, $f(\,j,n) := p(n) -2p(n-j) +p(n-2j)$ and give another estimate $f(\,j,n)$ . As applications these, a convexity property $p(n)$ as well giving new estimates k -rank $N_k(m,n)$ non- -ary partitions along with their differences.