Balanced polygamma function: Difference between revisions

Revision as of 19:39, 13 November 2013

Revision as of 04:45, 4 March 2014

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==Special values==

The balanced polygamma function can be expressed in a closed form at certain points:

* <math>\psi^{\left(-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^{\left(-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>

* <math>\psi^{\left(-3)}\left(,\frac12\right)=\frac1{16}\ln(2\pi)+\frac12\ln A+\frac{7\,\zeta(3)}{32\,\pi^2}</math>

* <math>\psi^{(-3)}(,1)=\frac14\ln(2\pi)+\ln A</math>

* <math>\psi^{(-3)}(,2)=\ln(2\pi)+2\ln A-\frac34</math>

==References==