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

## TL;DR

This paper proves a new formula for Stirling numbers of the second kind and applies it to derive algebraic topology results related to the unstable K-theory of stunted complex projective spaces.

## Contribution

It introduces a novel formula for Stirling numbers and applies it to establish isomorphisms in unstable K-theory of stunted complex projective spaces.

## Key findings

- Proved a new formula for Stirling numbers of the second kind.
- Showed divisibility properties of $k!S(n, k)$ for odd $n$ and even $k$.
- Derived isomorphisms of unstable $K^1$-groups for certain 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 $k+1\leq j\leq n$. Also, as an application to algebraic topology, some isomorphisms of unstable $K^1$-groups of stunted complex projective spaces are deduced.