I know the recurrence relation for the Polygamma function is


Does such a recurrence formula exist for negative integer $m$?

I am using the integral definition

$$\psi^{(-n)}(x)=\frac{1}{(x-2)!}\int_0^x (x-t)^{n-2}\ln(\Gamma(t))dt$$ for $n$ a positive integer, which I assume is equal to the $(n-1)$th integral of $\ln{\Gamma(x)}$.

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    $\begingroup$ Well, what definition of "negapolygamma" are you using? $\endgroup$ – J. M. is a poor mathematician Aug 17 '17 at 22:59
  • $\begingroup$ Note: my answer has been updated to provide a more explicit recursive relation of the polygamma function on negative orders, using the provided definition. $\endgroup$ – Simply Beautiful Art Aug 14 '18 at 1:57

Using your definition, we hence have


where $H_n=\sum_{k=}^n\frac1k$ is the harmonic number.


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