On extended Eulerian numbers

Abstract

In this paper, we will study some properties of the extended Eulerian numbers $H(n,\lambda)$, with a parameter $\lambda$. In fact, for any integer $n$, we investigate the asymptotic behavior, find lower and upper bounds for $H(n,\lambda)$. As application, for a champion number $N$, we obtain asymptotic formulas, lower and upper bounds of the arithmetic functions $\omega(N)$ and $\Omega(N)$.