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Balanced polygamma function: Difference between revisions

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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
Sure Beae (talk | contribs)
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→Relations: Fixed the improper algebra. The duplication formula introduced in the paper cannot be used like it was.
 
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\psi^{(n)}(x)&=\psi(n,x) \qquad n\in\mathbb{N} \\
\Gamma(x)&=\exp\left( \psi(-1,x)+\tfrac12 \ln 2\pi \right)\\
\zeta(z, q)&=\frac{\Gamma (-1-z)}{\ln 2} \left(2^{-z} {\psi \leftGamma(z-1,\frac{q+1}{2}\right)+2^{-z} \psi \left(z - 1,\frac{ q}{2}\right)-\psi(z-1,q)\right)\\
\zeta'(-1,x)&=\psi(-2, x) + \frac{x^2}2 - \frac{x}2 + \frac1{12} \\
B_n(q) &= -\frac{\Gamma (n+1)}{\ln 2} \left(2^{n-1} \psi\left(-n,\frac{q+1}{2}\right)+2^{n-1} \psi\left(-n,\frac{q}{2}\right)-\psi(-n,q)\right)
\end{align}</math>
 
where {{math|''B<sub>n</sub>''(''q'')}} are the [[Bernoulli polynomials]]
 
:<math>K(z)=A \exp\left(\psi(-2,z)+\frac{z^2-z}{2}\right)</math>
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