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DLMF: §5.8 Infinite Products ‣ Properties ‣ Chapter 5 Gamma Function

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5 Gamma Function Properties 5.7 Series Expansions 5.9 Integral Representations

§5.8 Infinite Products

ⓘ

Keywords:: gamma function, infinite products
Notes:: For (5.8.1)–(5.8.3) see Olver (1997b, pp. 34 and 38). For (5.8.4)–(5.8.5) see Whittaker and Watson (1927, p. 238–239).
Permalink:: http://dlmf.nist.gov/5.8
See also:: Annotations for Ch.5

5.8.1		$\Gamma\left(z\right)=\lim_{k\to\infty}\frac{k!k^{z}}{z(z+1)\cdots(z+k)},$
$z\neq 0,-1,-2,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 6.1.2 Referenced by: §5.8 Permalink: http://dlmf.nist.gov/5.8.E1 Encodings: TeX, pMML, png See also: Annotations for §5.8 and Ch.5

5.8.2		$\frac{1}{\Gamma\left(z\right)}=ze^{\gamma z}\prod_{k=1}^{\infty}\left(1+\frac{% z}{k}\right)e^{-z/k},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\gamma$ : Euler’s constant, $\mathrm{e}$ : base of natural logarithm, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 6.1.3 Permalink: http://dlmf.nist.gov/5.8.E2 Encodings: TeX, pMML, png See also: Annotations for §5.8 and Ch.5

5.8.3		$\left\|\frac{\Gamma\left(x\right)}{\Gamma\left(x+\mathrm{i}y\right)}\right\|^{2}% =\prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right),$
$x\neq 0,-1,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{i}$ : imaginary unit, $k$ : nonnegative integer, $x$ : real variable and $y$ : real variable A&S Ref: 6.1.25 (where the formula for the reciprocal is given.) Referenced by: §5.8 Permalink: http://dlmf.nist.gov/5.8.E3 Encodings: TeX, pMML, png See also: Annotations for §5.8 and Ch.5

If

5.8.4		$\sum_{k=1}^{m}a_{k}=\sum_{k=1}^{m}b_{k},$
ⓘ Defines: $a_{k}$ : coefficient (locally) and $b_{k}$ : coefficient (locally) Symbols: $m$ : nonnegative integer and $k$ : nonnegative integer Referenced by: §5.8 Permalink: http://dlmf.nist.gov/5.8.E4 Encodings: TeX, pMML, png See also: Annotations for §5.8 and Ch.5

then

5.8.5		$\prod_{k=0}^{\infty}\frac{(a_{1}+k)(a_{2}+k)\cdots(a_{m}+k)}{(b_{1}+k)(b_{2}+k% )\cdots(b_{m}+k)}=\frac{\Gamma\left(b_{1}\right)\Gamma\left(b_{2}\right)\cdots% \Gamma\left(b_{m}\right)}{\Gamma\left(a_{1}\right)\Gamma\left(a_{2}\right)% \cdots\Gamma\left(a_{m}\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $m$ : nonnegative integer, $k$ : nonnegative integer, $a_{k}$ : coefficient and $b_{k}$ : coefficient Referenced by: §5.8 Permalink: http://dlmf.nist.gov/5.8.E5 Encodings: TeX, pMML, png See also: Annotations for §5.8 and Ch.5

provided that none of the $b_{k}$ is zero or a negative integer.

5.7 Series Expansions 5.9 Integral Representations

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