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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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.
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