Difference between revisions of "Barnes G at positive integer"

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$$G(n) = \left\{ \begin{array}{ll}
 
$$G(n) = \left\{ \begin{array}{ll}
 
0&\quad n=-1,-2,\ldots \\
 
0&\quad n=-1,-2,\ldots \\
\displaystyle\prod_{i=0}^{n-2} i!&\quad n=0,1,2,\ldots,
+
\displaystyle\prod_{k=0}^{n-2} k!&\quad n=0,1,2,\ldots,
 
\end{array} \right.$$
 
\end{array} \right.$$
 
where $G$ denotes the [[Barnes G]] function and $i!$ denotes the [[factorial]].
 
where $G$ denotes the [[Barnes G]] function and $i!$ denotes the [[factorial]].
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[[Category:Theorem]]
 
[[Category:Theorem]]
 +
[[Category:Unproven]]

Latest revision as of 12:52, 17 September 2016

Theorem

The following formula holds: $$G(n) = \left\{ \begin{array}{ll} 0&\quad n=-1,-2,\ldots \\ \displaystyle\prod_{k=0}^{n-2} k!&\quad n=0,1,2,\ldots, \end{array} \right.$$ where $G$ denotes the Barnes G function and $i!$ denotes the factorial.

Proof

References