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* <math>K(z)=A e^{\psi(-2,z)+\frac{z^2-z}{2}}</math>
where ''K''(''z'') is [[K-function]] and A is the [[Glaisher constant]].
==Special values==
The balanced polygamma function can be expressed in a closed form at certain points:
* <math>\psi^{(-2)}\left(\frac14\right)=\frac18\ln(2\pi)+\frac98\ln A+\frac{G}{4\pi},</math> where <math>A</math> is the [[Glaisher constant]] and <math>G</math> is the [[Catalan constant]].
* <math>\psi^{(-2)}\left(\frac12\right)=\frac14\ln\pi+\frac32\ln A+\frac5{24}\ln2</math>
* <math>\psi^{(-2)}(1)=\frac12\ln(2\pi)</math>
* <math>\psi^{(-2)}(2)=\ln(2\pi)-1</math>
==References==
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