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:

    “A sophistical rhetorician, inebriated with the exuberance of his own verbosity, and gifted with an egotistical imagination that can at all times command an interminable and inconsistent series of arguments to malign an opponent and to glorify himself.”
    —Benjamin Disraeli (1804–1881)

    “So, if we must give a general formula applicable to all kinds of soul, we must describe it as the first actuality [entelechy] of a natural organized body.”
    —Aristotle (384–323 B.C.)