## INVERSE POLYGONAL NUMBERS SERIES-Notes

19 03 2009

The final result, in the preceeding post, can not be derived from a telescoping series [3], if $\displaystyle k$ is not integer (See comments at reference [1]).

$\displaystyle \sum_{n=1}^\infty \frac{1}{n(n+k)}=\frac{H_k}{k}$

This lack of generality, can be avoided, if we consider a more general definition for the Harmonic Numbers [4], extended to the complex plane, using the function:

$\displaystyle H_z=\gamma+\psi_0(z+1)$

Where $\displaystyle \psi_0 \;$ is the so called digamma function, and $\displaystyle \;\gamma\;$ is the Euler-Mascheroni constant.

If you take a look at the expresion (15), in the reference [2] : We can find that one asymptotic expansion for the digamma function is:

$\displaystyle \psi_0(k+1)=-\gamma+\sum_{n=1}^\infty{\frac{k}{n(n+k)}}$

This is why the Polygonal Numbers Series sum is working:

$\displaystyle H_k=\gamma-\gamma+ \sum_{n=1}^\infty{\frac{k}{n(n+k)}}$

$\displaystyle \frac{H_k}{k}=\sum_{n=1}^\infty{\frac{1}{n(n+k)}}=\frac{\gamma+\psi_0(k+1)}{k}$

And the polygonal numbers infinite sum, can be expressed (if $\displaystyle \;s\neq4\;$) as:

$\displaystyle S_{\infty}(s)=\frac{2}{4-s}*(\gamma+\psi_{0}\left(\frac{2}{s-2}\right))$

This expresion works for all $\displaystyle s>2$, as well as for all nonreal $\displaystyle s$, It also works for all $\displaystyle s<2$, except if $\displaystyle s<2$, and $\displaystyle s$ is $\displaystyle \;\;0, 1, 4/3, 6/4, 8/5, 10/6, ... \;$, because $\displaystyle \;\psi_0\;$ is not defined for negative integers (See reference) [1]

References:

[1]-Charles R Greathouse IV – Comments @ My Math Forum Inverse Polygonal Series
[2]-Weisstein, Eric W. “Digamma Function.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/DigammaFunction.html
[3]-Telescoping Series @ Wikipedia Telescoping Series
[4]-Sondow, Jonathan and Weisstein, Eric W. “Harmonic Number.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HarmonicNumber.html
[5]-Photo Martin Gardner, Mathematical Games, Scientific American, 211(5):126-133, taken from http://bit-player.org/2007/hung-over