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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.15 Sums

8.15.1 γ(a,λx)=λak=0γ(a+k,x)(1-λ)kk!.
8.15.2 ak=1(e2πik(z+h)(2πik)a+1Γ(a,2πikz)+e-2πik(z+h)(-2πik)a+1Γ(a,-2πikz))=ζ(-a,z+h)+za+1a+1+(h-12)za,
h[0,1].

For the Hurwitz zeta function ζ(s,a) see §25.11(i). For other infinite series whose terms include incomplete gamma functions, see Nemes (2017a), Reynolds and Stauffer (2021), and Prudnikov et al. (1986b, §5.2).