§8.7 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).
Keywords:: Bessel functions, Laguerre polynomials, incomplete gamma functions, modified spherical Bessel functions
Referenced by:: §8.25(i)
Permalink:: http://dlmf.nist.gov/8.7

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 $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)=e^{{-z}}\sum _{{k=0}}^{\infty}\frac{z^{k}}{\mathop{\Gamma\/}\nolimits\!\left(a+k+1\right)}=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}\sum _{{k=0}}^{\infty}\frac{(-z)^{k}}{k!(a+k)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : 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

8.7.2 $\mathop{\gamma\/}\nolimits\!\left(a,x+y\right)-\mathop{\gamma\/}\nolimits\!\left(a,x\right)=\mathop{\Gamma\/}\nolimits\!\left(a,x\right)-\mathop{\Gamma\/}\nolimits\!\left(a,x+y\right)=e^{{-x}}x^{{a-1}}\sum _{{n=0}}^{\infty}\frac{\left(1-a\right)_{{n}}}{(-x)^{n}}(1-e^{{-y}}e_{n}(y)),$ $|y|<|x|$ .

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : 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

8.7.3 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=\mathop{\Gamma\/}\nolimits\!\left(a\right)-\sum _{{k=0}}^{\infty}\frac{(-1)^{k}z^{{a+k}}}{k!(a+k)}=\mathop{\Gamma\/}\nolimits\!\left(a\right)\left(1-z^{a}e^{{-z}}\sum _{{k=0}}^{\infty}\frac{z^{k}}{\mathop{\Gamma\/}\nolimits\!\left(a+k+1\right)}\right),$ $a\neq 0,-1,-2,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : 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

8.7.4 $\mathop{\gamma\/}\nolimits\!\left(a,x\right)=\mathop{\Gamma\/}\nolimits\!\left(a\right)x^{{\frac{1}{2}a}}e^{{-x}}\sum _{{n=0}}^{\infty}e_{n}(-1)x^{{\frac{1}{2}n}}\mathop{I_{{n+a}}\/}\nolimits\!\left(\textstyle 2x^{{1/2}}\right),$ $a\neq 0,-1,-2,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $x$ : real variable, $a$ : parameter, $n$ : nonnegative integer and $e_{n}(z)$ : functions
Permalink:: http://dlmf.nist.gov/8.7.E4
Encodings:: TeX, pMML, png

8.7.5 $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)=e^{{-\frac{1}{2}z}}\sum _{{n=0}}^{\infty}\frac{\left(1-a\right)_{{n}}}{\mathop{\Gamma\/}\nolimits\!\left(n+a+1\right)}{\left(2n+1\right)}\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(\tfrac{1}{2}z\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : 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

8.7.6 $\mathop{\Gamma\/}\nolimits\!\left(a,x\right)=x^{a}e^{{-x}}\sum _{{n=0}}^{\infty}\frac{\mathop{L^{{(a)}}_{{n}}\/}\nolimits\!\left(x\right)}{n+1},$ $x>0$ .

Symbols:: $e$ : base of exponential function, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $x$ : real variable, $a$ : parameter and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/8.7.E6
Encodings:: TeX, pMML, png

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).