# §8.15 Sums

 8.15.1 $\gamma\left(a,\lambda x\right)=\lambda^{a}\sum_{k=0}^{\infty}\gamma\left(a+k,x% \right)\frac{(1-\lambda)^{k}}{k!}.$
 8.15.2 $a\sum_{k=1}^{\infty}\left(\frac{{\mathrm{e}}^{2\pi\mathrm{i}k(z+h)}}{\left(2% \pi\mathrm{i}k\right)^{a+1}}\Gamma\left(a,2\pi\mathrm{i}kz\right)+\frac{{% \mathrm{e}}^{-2\pi\mathrm{i}k(z+h)}}{\left(-2\pi\mathrm{i}k\right)^{a+1}}% \Gamma\left(a,-2\pi\mathrm{i}kz\right)\right)=\zeta\left(-a,z+h\right)+\frac{z% ^{a+1}}{a+1}+\left(h-\tfrac{1}{2}\right)z^{a},$ $h\in[0,1]$. ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$: closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\Gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer Proof sketch: Start with $a<-1$ and $z>0$. For $\Gamma\left(a,\pm 2\pi\mathrm{i}kz\right)$ take integral representation (8.2.2) and use the substitution $t=\pm 2\pi\mathrm{i}kz(1+\tau)$. The sum and the integral can be interchanged, and the sum can be evaluated via (4.6.1). Use integration by parts. This will result in $\left(h-\tfrac{1}{2}\right)z^{a}$ plus two integrals with infinitely many poles. The residue theorem (§1.10(iv)) will give us an infinite series which can be identified via (25.11.1). For other values of $a$ and $z$ use analytic continuation. Referenced by: §25.11(x), §25.11(x), §8.15, Erratum (V1.1.3) for Additions Permalink: http://dlmf.nist.gov/8.15.E2 Encodings: TeX, pMML, png Addition (effective with 1.1.3): This equation was added. See also: Annotations for §8.15 and Ch.8

For the Hurwitz zeta function $\zeta\left(s,a\right)$ 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).