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Hi, I'm looking for a link to a derivation of some of the basic properties of Hadamard's Gamma function. For instance that it satisfies $H(x+1)=xH(x)+\frac{1}{\Gamma(1-x)}$ I've been looking on the internet and couldn't really find much accessible literature on it (or any at all for that matter!).

Tom.

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2 Answers

I assume you have already found this page? It seems that, from the definition $$ H(x) = \frac{\Psi(1-x/2)-\Psi(1/2-x/2)}{2\Gamma(1-x)} $$ the various properties you are interested in should be straightforward to prove.

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Yes, that's where I looked first. Sorry but how do they follow immediately from this formulation of H(x)?

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From the reflection formulas from $\Psi$. Both Maple and Mathematica immediately simplify $H(x+1)-x*H(x)$ to $1/\Gamma(1-x)$ when using that definition, by using those reflection formulas and basic properties of $\Gamma$. –  Jacques Carette Feb 27 '10 at 15:51
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@Tom: Please use the comment function to comment on other people's answers instead of adding your own answer. Even though you don't have the necessary reputation to comment in general, you're supposed to be able to comment on answers to your own question (tea.mathoverflow.net/discussion/50/…). –  Harald Hanche-Olsen Feb 27 '10 at 16:50
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@Harald Hanche-Olsen. The Tom who asked the question, and the Tom of the above answer, have different userids. Usual problems of not being a registered user, I would imagine. –  Regenbogen Feb 27 '10 at 18:20
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