2019/09/16 by DeFranco, Mario
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1909.06941
We define an absolutely convergent series for the upper incomplete Gamma function Γ(s,z) for z≥ 1 and s∈ 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 ∈ ℂ.