§8.7 Series Expansions

ⓘ

Keywords:: Bessel functions, Laguerre polynomials, expansions in series of, incomplete gamma functions, modified spherical Bessel functions, power-series expansions
Notes:: See Erdélyi et al. (1953b, p. 135 and §9.4). (8.7.3) follows from (8.7.1) and (8.2.3). For (8.7.5) use (8.5.2) and (13.11.1).
Referenced by:: §8.25(i)
Permalink:: http://dlmf.nist.gov/8.7
See also:: Annotations for Ch.8

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		$\gamma^{*}\left(a,z\right)=e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left(a% +k+1\right)}=\frac{1}{\Gamma\left(a\right)}\sum_{k=0}^{\infty}\frac{(-z)^{k}}{% k!(a+k)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : 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		$\gamma\left(a,x+y\right)-\gamma\left(a,x\right)=\Gamma\left(a,x\right)-\Gamma% \left(a,x+y\right)=e^{-x}x^{a-1}\sum_{n=0}^{\infty}\frac{{\left(1-a\right)_{n}% }}{(-x)^{n}}(1-e^{-y}e_{n}(y)),$
$\|y\|<\|x\|$ .
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\gamma\left(\NVar{a},\NVar{z}\right)$ : 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		$\Gamma\left(a,z\right)=\Gamma\left(a\right)-\sum_{k=0}^{\infty}\frac{(-1)^{k}z% ^{a+k}}{k!(a+k)}=\Gamma\left(a\right)\left(1-z^{a}e^{-z}\sum_{k=0}^{\infty}% \frac{z^{k}}{\Gamma\left(a+k+1\right)}\right),$
$a\neq 0,-1,-2,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\Gamma\left(\NVar{a},\NVar{z}\right)$ : 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		$\gamma\left(a,x\right)=\Gamma\left(a\right)x^{\frac{1}{2}a}e^{-x}\sum_{n=0}^{% \infty}e_{n}(-1)x^{\frac{1}{2}n}I_{n+a}\left(\textstyle 2x^{1/2}\right),$
$a\neq 0,-1,-2,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		$\gamma^{*}\left(a,z\right)=e^{-\frac{1}{2}z}\sum_{n=0}^{\infty}\frac{{\left(1-% a\right)_{n}}}{\Gamma\left(n+a+1\right)}{\left(2n+1\right)}{\mathsf{i}^{(1)}_{% n}}\left(\tfrac{1}{2}z\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : 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		$\Gamma\left(a,x\right)=x^{a}e^{-x}\sum_{n=0}^{\infty}\frac{L^{(a)}_{n}\left(x% \right)}{n+1},$
$x>0$ , $\Re a<\frac{1}{2}$ .
ⓘ 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).