# partial fraction series for digamma function

###### Theorem 1
 $\psi(z)=-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{z+k}% \right)=-\gamma+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{z+k}\right)$

 $\Gamma(z)=\frac{e^{-\gamma z}}{z}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right% )^{-1}e^{z/k},$

so

 $\ln\Gamma(z)=-\gamma z-\ln z+\sum_{k=1}^{\infty}\left(-\ln\left(1+\frac{z}{k}% \right)+\frac{z}{k}\right)$

and thus, taking derivatives,

 $\psi(z)=-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\left(-\frac{1/k}{1+\frac{z}{k}% }+\frac{1}{k}\right)=-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\left(\frac{1}{k}-% \frac{1}{z+k}\right)$

The second formula follows after rearranging terms (the rearrangement is legal since we are simply exchanging adjacent terms, so partial sums remain the same).

Title partial fraction series for digamma function PartialFractionSeriesForDigammaFunction 2013-03-22 16:23:40 2013-03-22 16:23:40 rm50 (10146) rm50 (10146) 6 rm50 (10146) Theorem msc 33B15 msc 30D30