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For what values of $\nu$ does the Riemann-Liouville differintegral $_{-\infty}D_{z}^\nu$ of the digamma function $\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$ exist, with $c=-\infty$? All I've got so far is that the derivatives exist, i.e. the differintegrals when $\mathrm{Re}(\nu)>0 $.

Many thanks for any help with this!

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Can you show your work? –  Mhenni Benghorbal Oct 12 '12 at 22:47
    
The proof that the derivatives exist is a bit long to include here! :-) But I've now sorted this out: the definition of fractional derivatives assumes the existence of fractional integrals with $-1<\mathrm{Re}(\nu)<0$, so we must be able to extend to $\mathrm{Re}(\nu)>-1$, which was all I needed. –  Harry Macpherson Oct 14 '12 at 18:35
    
Have you by any chance seen the paper by Moll and Espinosa? That might give you something to start with... still, the poles of the polygamma functions in the left half-plane has me skeptical that the beasts are well-defined for noninteger $\nu$ and lower limit $-\infty$... –  J. M. Nov 25 '12 at 10:41
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