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.

Read more about this topic:  Digamma Function

Famous quotes containing the words series and/or formula:

    “Personality is an unbroken series of successful gestures.”
    —F. Scott Fitzgerald (1896–1940)

    “Ideals possess the strange quality that if they were completely realized they would turn into nonsense. One could easily follow a commandment such as “Thou shalt not kill” to the point of dying of starvation; and I might establish the formula that for the proper functioning of the mesh of our ideals, as in the case of a strainer, the holes are just as important as the mesh.”
    —Robert Musil (1880–1942)