§8.7 Series Expansions

For the functions $e_{n}(z)$, ${\mathsf{i}^{(1)}_{n}}\left(z\right)$, and $L^{(\alpha)}_{n}\left(x\right)$ see (8.4.11), §§10.47(ii), and 18.3, respectively.

 8.7.1 $\gamma^{*}\left(a,z\right)=e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left(a% +k+1\right)}=\frac{1}{\Gamma\left(a\right)}\sum_{k=0}^{\infty}\frac{(-z)^{k}}{% k!(a+k)}.$
 8.7.2 $\gamma\left(a,x+y\right)-\gamma\left(a,x\right)=\Gamma\left(a,x\right)-\Gamma% \left(a,x+y\right)=e^{-x}x^{a-1}\sum_{n=0}^{\infty}\frac{{\left(1-a\right)_{n}% }}{(-x)^{n}}(1-e^{-y}e_{n}(y)),$ $|y|<|x|$.
 8.7.3 $\Gamma\left(a,z\right)=\Gamma\left(a\right)-\sum_{k=0}^{\infty}\frac{(-1)^{k}z% ^{a+k}}{k!(a+k)}=\Gamma\left(a\right)\left(1-z^{a}e^{-z}\sum_{k=0}^{\infty}% \frac{z^{k}}{\Gamma\left(a+k+1\right)}\right),$ $a\neq 0,-1,-2,\dots$.
 8.7.4 $\gamma\left(a,x\right)=\Gamma\left(a\right)x^{\frac{1}{2}a}e^{-x}\sum_{n=0}^{% \infty}e_{n}(-1)x^{\frac{1}{2}n}I_{n+a}\left(\textstyle 2x^{1/2}\right),$ $a\neq 0,-1,-2,\dots$.
 8.7.5 $\gamma^{*}\left(a,z\right)=e^{-\frac{1}{2}z}\sum_{n=0}^{\infty}\frac{{\left(1-% a\right)_{n}}}{\Gamma\left(n+a+1\right)}{\left(2n+1\right)}{\mathsf{i}^{(1)}_{% n}}\left(\tfrac{1}{2}z\right).$
 8.7.6 $\Gamma\left(a,x\right)=x^{a}e^{-x}\sum_{n=0}^{\infty}\frac{L^{(a)}_{n}\left(x% \right)}{n+1},$ $x>0$, $\Re a<\frac{1}{2}$. ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $\Re$: real part, $x$: real variable, $z$: complex variable, $a$: parameter and $n$: nonnegative integer Proof sketch: Integrate (18.12.13) from $z=0$ to $z=1$ and for the integral on the left-hand side use the substitution $z=\frac{t}{1+t}$. The resulting integral is (8.6.5). The constraint $\Re a<\tfrac{1}{2}$ follows from (18.15.14). Referenced by: Erratum (V1.1.8) for Equation (8.7.6) Permalink: http://dlmf.nist.gov/8.7.E6 Encodings: TeX, pMML, png Addition (effective with 1.1.8): The constraint was updated to include $\Re a<\frac{1}{2}$. Suggested 2022-10-14 by Walter Gautschi See also: Annotations for §8.7 and Ch.8

For an expansion for $\gamma\left(a,ix\right)$ in series of Bessel functions $J_{n}\left(x\right)$ that converges rapidly when $a>0$ and $x$ ($\geq 0$) is small or moderate in magnitude see Barakat (1961).