8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions8.14 Integrals8.16 Generalizations

§8.15 Sums

ⓘ

Keywords:

incomplete gamma functions, sums

Notes:

See Tricomi (1950b).

Permalink:

http://dlmf.nist.gov/8.15

Addition (effective with 1.1.3):

Equation (8.15.2) was added.

See also:

Annotations for Ch.8

8.15.1		$$\gamma (a,\lambda x)={\lambda }^a\sum _{k=0}^{\mathrm{\infty }}\gamma (a+k,x)\frac{{(1-\lambda )}^k}{k!}.$$
ⓘ Symbols: $!$: factorial (as in $n!$), $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter and $k$: nonnegative integer Permalink: http://dlmf.nist.gov/8.15.E1 Encodings: TeX, pMML, png See also: Annotations for §8.15 and Ch.8

8.15.2		$$\begin{array}{l}a\sum _{k=1}^{\mathrm{\infty }}\left(\frac{{\mathrm{e}}^{2\pi \mathrm{i}k(z+h)}}{{\left(2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,2\pi \mathrm{i}kz)+\frac{{\mathrm{e}}^{-2\pi \mathrm{i}k(z+h)}}{{\left(-2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,-2\pi \mathrm{i}kz)\right)\\ =\zeta (-a,z+h)+\frac{z^{a+1}}{a+1}+\left(h-\frac{1}{2}\right)z^a,\end{array}$$
$h\in [0,1]$.
ⓘ Symbols: $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer Proof sketch: Start with and $z>0$. For $\mathrm{\Gamma }(a,\pm 2\pi \mathrm{i}kz)$ 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 $\left(h-\frac{1}{2}\right)z^a$ 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 $\zeta (s,a)$ 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).

8.14 Integrals8.16 Generalizations

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