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It is well known that $\psi\left( x+N\right) =\psi\left( x\right) +\sum_{k=0}^{N-1}\frac{1}{x+k}$.

Is there a recurrence formula for $\psi\left( x+\frac{p}{q}\right) $ where $\frac{p}{q}$ is rational number

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  • $\begingroup$ I guess you had a look at the wikipedia page en.wikipedia.org/wiki/Digamma_function#Gaussian_sum. There are some nice formulas, but maybe not quite what you're looking for. $\endgroup$ Dec 19, 2013 at 7:23
  • $\begingroup$ Your first formula comes from $\Gamma(x+1)=x\Gamma(x)$. So maybe your question is related to whether there is a formula $\Gamma(x+1/q) = \dots$. $\endgroup$ Dec 19, 2013 at 15:17

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