A formula on Stirling numbers of the second kind and its application to the unstable K-theory of stunted complex projective spaces

Abstract

A formula on Stirling numbers of the second kind $S(n,k)$ is proved. As a corollary, for odd n and even k, it is shown that $k!S(n,k)$ is a positive multiple of the greatest common divisor of $j!S(n,j)$ for $k+1\le j\le n$. Also, as an application to algebraic topology, some isomorphisms of unstable $K^1$-groups of stunted complex projective spaces are deduced.