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:

    “Through a series of gradual power losses, the modern parent is in danger of losing sight of her own child, as well as her own vision and style. It’s a very big price to pay emotionally. Too bad it’s often accompanied by an equally huge price financially.”
    —Sonia Taitz (20th century)

    “My formula for greatness in human beings is amor fati: that one wants to change nothing, neither forwards, nor backwards, nor in all eternity. Not merely to endure necessity, still less to hide it—all idealism is mendacity in the face of necessity—but rather to love it.”
    —Friedrich Nietzsche (1844–1900)