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Citation bot (talk | contribs) Add: doi, pages. Removed parameters. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | Suggested by Abductive | Category:Gamma and related functions | #UCB_Category 30/38 |
The original special values listed here for the generalized polygamma function are not from the balanced version of the function. They were taken from repeated integration from 0 to x of the log-gamma function, which is not balanced. The new values were calculated using the Hurwitz zeta representation given in this article which was verified to be balanced using Desmos: https://www.desmos.com/calculator/aw9ugcyfvs |
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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
\psi\left(-2,\tfrac12\right)&=\
\psi\left(-3,\tfrac12\right)&=
\psi(-2,1)&=
\psi(-3,1)&=\
\psi(-2,2)&=-\ln
\psi(-3,2)&=\
==References==
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