On a series for the upper incomplete Gamma function

TL;DR

This paper introduces a new absolutely convergent series for the upper incomplete Gamma function using Stirling number-based polynomials, and applies it to derive a formula for the Riemann xi function valid for all complex s.

Contribution

It presents a novel series expansion for the upper incomplete Gamma function and demonstrates its application to a general formula for the Riemann xi function.

Findings

Series converges absolutely for z ≥ 1 and complex s.

Polynomials used have proven positive coefficients.

Derived a new formula for the Riemann xi 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 $s \in \mathbb{C}$.