# §8.7 Series Expansions

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

 8.7.1 $\mathop{\gamma^{*}\/}\nolimits\!\left(a,z\right)=e^{-z}\sum_{k=0}^{\infty}% \frac{z^{k}}{\mathop{\Gamma\/}\nolimits\!\left(a+k+1\right)}=\frac{1}{\mathop{% \Gamma\/}\nolimits\!\left(a\right)}\sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!(a+k)}.$
 8.7.2 $\mathop{\gamma\/}\nolimits\!\left(a,x+y\right)-\mathop{\gamma\/}\nolimits\!% \left(a,x\right)=\mathop{\Gamma\/}\nolimits\!\left(a,x\right)-\mathop{\Gamma\/% }\nolimits\!\left(a,x+y\right)=e^{-x}x^{a-1}\sum_{n=0}^{\infty}\frac{{\left(1-% a\right)_{n}}}{(-x)^{n}}(1-e^{-y}e_{n}(y)),$ $|y|<|x|$.
 8.7.3 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=\mathop{\Gamma\/}\nolimits\!\left% (a\right)-\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{a+k}}{k!(a+k)}=\mathop{\Gamma\/}% \nolimits\!\left(a\right)\left(1-z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{% \mathop{\Gamma\/}\nolimits\!\left(a+k+1\right)}\right),$ $a\neq 0,-1,-2,\dots$.
 8.7.4 $\mathop{\gamma\/}\nolimits\!\left(a,x\right)=\mathop{\Gamma\/}\nolimits\!\left% (a\right)x^{\frac{1}{2}a}e^{-x}\sum_{n=0}^{\infty}e_{n}(-1)x^{\frac{1}{2}n}% \mathop{I_{n+a}\/}\nolimits\!\left(\textstyle 2x^{1/2}\right),$ $a\neq 0,-1,-2,\dots$.
 8.7.5 $\mathop{\gamma^{*}\/}\nolimits\!\left(a,z\right)=e^{-\frac{1}{2}z}\sum_{n=0}^{% \infty}\frac{{\left(1-a\right)_{n}}}{\mathop{\Gamma\/}\nolimits\!\left(n+a+1% \right)}{\left(2n+1\right)}\mathop{{\mathsf{i}^{(1)}_{n}}\/}\nolimits\!\left(% \tfrac{1}{2}z\right).$
 8.7.6 $\mathop{\Gamma\/}\nolimits\!\left(a,x\right)=x^{a}e^{-x}\sum_{n=0}^{\infty}% \frac{\mathop{L^{(a)}_{n}\/}\nolimits\!\left(x\right)}{n+1},$ $x>0$.

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