§5.8 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).
Keywords:: gamma function, infinite products
Permalink:: http://dlmf.nist.gov/5.8

5.8.1 $\mathop{\Gamma\/}\nolimits\!\left(z\right)=\lim _{{k\to\infty}}\frac{k!k^{z}}{z(z+1)\cdots(z+k)},$ $z\neq 0,-1,-2,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $!$ : $n!$ : factorial, $k$ : 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

5.8.2 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(z\right)}=ze^{{\EulerConstant z}}\prod _{{k=1}}^{\infty}\left(1+\frac{z}{k}\right)e^{{-z/k}},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\EulerConstant$ : Euler’s constant, $e$ : base of exponential function, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.1.3
Permalink:: http://dlmf.nist.gov/5.8.E2
Encodings:: TeX, pMML, png

5.8.3 $\left|\frac{\mathop{\Gamma\/}\nolimits\!\left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(x+iy\right)}\right|^{2}=\prod _{{k=0}}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right),$ $x\neq 0,-1,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $k$ : 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

5.8.4 $\sum _{{k=1}}^{m}a_{k}=\sum _{{k=1}}^{m}b_{k},$

Symbols:: $m$ : nonnegative integer, $k$ : nonnegative integer, $a_{k}$ : coefficient and $b_{k}$ : coefficient
Referenced by:: §5.8
Permalink:: http://dlmf.nist.gov/5.8.E4
Encodings:: TeX, pMML, png

then

5.8.5 $\prod _{{k=0}}^{\infty}\frac{(a_{1}+k)(a_{2}+k)\cdots(a_{m}+k)}{(b_{1}+k)(b_{2}+k)\cdots(b_{m}+k)}=\frac{\mathop{\Gamma\/}\nolimits\!\left(b_{1}\right)\mathop{\Gamma\/}\nolimits\!\left(b_{2}\right)\cdots\mathop{\Gamma\/}\nolimits\!\left(b_{m}\right)}{\mathop{\Gamma\/}\nolimits\!\left(a_{1}\right)\mathop{\Gamma\/}\nolimits\!\left(a_{2}\right)\cdots\mathop{\Gamma\/}\nolimits\!\left(a_{m}\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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

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