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The balanced polygamma function can be expressed in a closed form at certain points (where {{mvar|A}} is the [[Glaisher constant]] and {{mvar|G}} is the [[Catalan constant]]):
:<math>\begin{align}
\psi\left(-2,\tfrac14\right)&=\tfrac18\ln 2\pi+\tfrac98\ln A+\frac{G}{4\pi} && \\
\psi\left(-2,\tfrac12\right)&=\tfrac14\ln\pi+\tfrac32\ln A+\tfrac5{24}\ln2 & \\
\psi\left(-3,\tfrac12\right)&=\tfrac1{16}\ln 2\pi+\tfrac12\ln A+\frac{7\zeta(3)}{32\pi^2}\\
\psi(-2,1)&=\tfrac12\ln 2\pi &\\
\psi(-3,1)&=\tfrac14\ln 2\pi+\ln A\\
\psi(-2,2)&=\ln 2\pi-1 &\\
\psi(-3,2)&=\ln 2\pi+2\ln A-\tfrac34 \\\end{align}</math>
==References==
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