On a series for the upper incomplete Gamma function

Abstract

We define an absolutely convergent series for the upper incomplete Gamma function $\Gamma(s,z)$ for $z\geq 1$ and $s\in \mathbb{C}$. We express this series using certain polynomials which we define using the Stirling numbers of the first kind. We prove that these polynomials have positive coefficients by defining a three-parameter family of integers and certain linear operators on vector spaces of polynomials. We then apply this series to obtain a formula for the Riemann xi function valid at any $s \in \mathbb{C}$.