DLMF: GA.8 Infinite Products

§GA.8. Infinite Products

Sources:

Olver(1997)(Chapter 2, §1), Whittaker and Watson(1927)(§12.13).

Notes:

GA.8.1

GA.8.3

Olver(1997)(pp. 34 and 38)

GA.8.4

GA.8.5

Whittaker and Watson(1927)(p. 238–239)

Keywords:

gamma function, infinite product

GA.8.1 $\Gamma\!\left(z\right)=\lim _{{k\to\infty}}\frac{k!k^{z}}{z(z+1)\cdots(z+k)},$ $z\neq 0,-1,-2,\dots$ ,

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.1.2
Referenced by:: §GA.8
Encodings:: pMathML, png, TeX

GA.8.2 $\frac{1}{\Gamma\!\left(z\right)}=ze^{{\EulerConstant z}}\prod _{{k=1}}^{\infty}\left(1+\frac{z}{k}\right)e^{{-z/k}},$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $\EulerConstant$ : Euler's constant, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.1.3
Encodings:: pMathML, png, TeX

GA.8.3 $\left|\frac{\Gamma\!\left(x\right)}{\Gamma\!\left(x+iy\right)}\right|^{2}=\prod _{{k=0}}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right),$ $x\neq 0,-1,\dots$ .

Notations:: $\Gamma\!\left(z\right)$ : 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:: §GA.8
Encodings:: pMathML, png, TeX

GA.8.4 $\sum _{{k=1}}^{m}a_{k}=\sum _{{k=1}}^{m}b_{k},$

Notations:: $m$ : nonnegative integer, $k$ : nonnegative integer, $a_{k}$ : coefficient and $b_{k}$ : coefficient
Referenced by:: §GA.8
Encodings:: pMathML, pMathML, pMathML, png, png, png, TeX, TeX, TeX

then

GA.8.5 $\prod _{{k=0}}^{\infty}\frac{(a_{1}+k)(a_{2}+k)\cdots(a_{m}+k)}{(b_{1}+k)(b_{2}+k)\cdots(b_{m}+k)}=\frac{\Gamma\!\left(b_{1}\right)\Gamma\!\left(b_{2}\right)\cdots\Gamma\!\left(b_{m}\right)}{\Gamma\!\left(a_{1}\right)\Gamma\!\left(a_{2}\right)\cdots\Gamma\!\left(a_{m}\right)},$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $m$ : nonnegative integer, $k$ : nonnegative integer, $a_{k}$ : coefficient and $b_{k}$ : coefficient
Referenced by:: §GA.8
Encodings:: pMathML, png, TeX

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