Prof. Putinar's Infinite Product: Ben Orin's Addition

[benorin]

Prof. Putinar,

Guillera & Sondow gave

$$e^{x}=\prod_{n=1}^{\infty}\left(\prod_{k=1}^{n} (1+kx)^{(-1)^{k+1}\left(\begin{array}{c}n\\k\end{array}\right)} \right) ^{\frac{1}{n}}$$​

for $$x\geq 0$$, to which I add

$$\boxed{\frac{e^{u}\Gamma (u)}{\sqrt{2\pi e}}=\prod_{n=0}^{\infty}\left(\prod_{k=0}^{n} (k+u)^{(-1)^{k}\left(\begin{array}{c}n\\k\end{array}\right) (k+u)} \right) ^{\frac{1}{n+1}}}$$​

for $$\mbox{Re} <u>\geq 0</u>$$.

-Ben Orin

 

[Zurtex]

Cool.

 

[benorin]

The paper of Guillera & Sondow cited above is entitled http://arxiv.org/PS_cache/math/pdf/0506/0506319.pdf and their result (the infinite product for $$e^x$$) is equation (58) on pg. 18.

 

[benorin]

An Infinite Product for e^x

Edit: Fixing my post for TeX and updating link to paper.

Prof. Putinar,

Guillera & Sondow¹ gave $e^{x} = \prod_{n=1}^{\infty}\left( \prod_{k=1}^{n} (kx+1) ^{(-1)^{k+1} \left( \begin{array}{c}n\\k\end{array}\right) } \right) ^{\frac{1}{n}}\mbox{ for }x\geq 0,$
to it's company I add $\frac{e^{u}\Gamma (u)}{\sqrt{2\pi e}}=\prod_{n=0}^{\infty}\left(\prod_{k=0}^{n}(k+u)^{<br /> (-1)^{k}\left(\begin{array}{c}n\\k\end{array}\right)(k+u)}\right)^{\frac{1}{n+1}}\mbox{ for }\mbox{Re} <u>\geq 0.</u>$



-Ben Orin

benorin@umail.ucsb.edu

¹ The infinite product for $e^{x}$ is (60) on pg. 20 of http://arxiv.org/abs/math/0506319" .)