§8.7 Series Expansions

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

8.7.1		$${\gamma }^{*}(a,z)={\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)}=\frac{1}{\mathrm{\Gamma }\left(a\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-z)}^k}{k!(a+k)}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), ${\gamma }^{*}(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer A&S Ref: 6.5.29 Referenced by: §8.11(ii), §8.14, §8.6(i), §8.7 Permalink: http://dlmf.nist.gov/8.7.E1 Encodings: TeX, pMML, png See also: Annotations for §8.7 and Ch.8

8.7.2		$$\gamma (a,x+y)-\gamma (a,x)=\mathrm{\Gamma }(a,x)-\mathrm{\Gamma }(a,x+y)={\mathrm{e}}^{-x}{x}^{a-1}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_n}{{(-x)}^n}(1-{\mathrm{e}}^{-y}{e}_n(y)),$$
.
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter, $n$: nonnegative integer and $e_n(z)$: functions A&S Ref: 6.5.30 Permalink: http://dlmf.nist.gov/8.7.E2 Encodings: TeX, pMML, png See also: Annotations for §8.7 and Ch.8

8.7.3		$$\mathrm{\Gamma }(a,z)=\mathrm{\Gamma }\left(a\right)-\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{z}^{a+k}}{k!(a+k)}=\mathrm{\Gamma }\left(a\right)\left(1-z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)}\right),$$
$a\ne 0,-1,-2,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer Referenced by: §8.14, §8.19(iv), §8.4, §8.7 Permalink: http://dlmf.nist.gov/8.7.E3 Encodings: TeX, pMML, png See also: Annotations for §8.7 and Ch.8

8.7.4		$$\gamma (a,x)=\mathrm{\Gamma }\left(a\right)x^{\frac{1}{2}a}{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}{e}_n(-1)x^{\frac{1}{2}n}{I}_{n+a}\left(2x^{1/2}\right),$$
$a\ne 0,-1,-2,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $a$: parameter, $n$: nonnegative integer and $e_n(z)$: functions Permalink: http://dlmf.nist.gov/8.7.E4 Encodings: TeX, pMML, png See also: Annotations for §8.7 and Ch.8

8.7.5		$${\gamma }^{*}(a,z)={\mathrm{e}}^{-\frac{1}{2}z}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_n}{\mathrm{\Gamma }\left(n+a+1\right)}\left(2n+1\right){\mathsf{i}}_n^{(1)}\left(\frac{1}{2}z\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, ${\gamma }^{*}(a,z)$: incomplete gamma function, ${\mathsf{i}}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable, $a$: parameter and $n$: nonnegative integer Referenced by: §8.7 Permalink: http://dlmf.nist.gov/8.7.E5 Encodings: TeX, pMML, png See also: Annotations for §8.7 and Ch.8

8.7.6		$$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$
$x>0$.
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/8.7.E6 Encodings: TeX, pMML, png See also: Annotations for §8.7 and Ch.8

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