Digamma Function - Series Formula

Series Formula

Digamma can be computed in the complex plane outside negative integers (Abramowitz and Stegun 6.3.16), using

or

This can be utilized to evaluate infinite sums of rational functions, i.e., where p(n) and q(n) are polynomials of n.

Performing partial fraction on un in the complex field, in the case when all roots of q(n) are simple roots,

For the series to converge,

or otherwise the series will be greater than harmonic series and thus diverges.

Hence

and

=\sum_{k=1}^{m}\left(a_{k}\sum_{n=0}^{\infty}\left(\frac{1}{n+b_{k}}-\frac{1}{n+1}\right)\right)=-\sum_{k=1}^{m}a_{k}\left(\psi(b_{k})+\gamma\right)=-\sum_{k=1}^{m}a_{k}\psi(b_{k}).

With the series expansion of higher rank polygamma function a generalized formula can be given as

provided the series on the left converges.

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