The Barnes' G-Function, and related higher functions

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In summary, the Barnes' G-function is a mathematical function used to generalize factorials and gamma functions. It has many applications in number theory, combinatorics, and statistical mechanics. The Barnes' G-function has several related higher functions, such as the multi-factorial function and the double gamma function, which extend its usefulness in various fields of mathematics. These functions have been studied extensively and have been shown to have connections to other areas of mathematics, making them an important tool in mathematical research.
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Random Variable said:
I haven't thought about it much, but perhaps it can be derived by using the multiplication formula for the Hurwitz zeta function.
I think you might be right there, RV... There's nothing scientific about it, obviously, but I 'feel' like I'm not far off it. At least that's the hope: just a bit more gruntwork and I'll accidentally stumble across it when least expected.

It's quite a formidable formula, though... (Headbang)I suspect the relation \(\displaystyle \log G(1+z)-z\log \Gamma(z)= \zeta ' (-1)-\zeta ' (-1,z)\) might also offer a decent starting point.
 

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