Balanced polygamma function: Difference between revisions

Revision as of 19:43, 27 December 2016

Revision as of 19:44, 27 December 2016

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\psi^{(n)}(x)&=\psi(n,x) \qquad n\in\mathbb{N} \\[8px]

\Gamma(x)&=\exp\left( \psi(-1,x)+\tfrac12 \ln 2\pi \right)\\[8px]

\zeta(z,q)&=\frac{\Gamma (1-z)}{\ln 2} \left(2^{-z} \psi \left(z-1,\tfrac12 frac{q+\tfrac{1}{2}\right)+2^{-z} \psi \left(z-1,\tfrac12 frac{q}{2}\right)-\psi(z-1,q)\right) \\[8px]

\zeta'(-1,x)&=\psi(-2, x) + \frac{x^2}2 - \frac{x}2 + \frac1{12} \\[8px]

B_n(q) &= -\frac{\Gamma (n+1)}{\ln 2} \left(2^{n-1} \psi\left(-n,\tfrac12 frac{q+\tfrac{1}{2}\right)+2^{n-1} \psi\left(-n,\tfrac12 frac{q}{2}\right)-\psi(-n,q)\right)

\end{align}</math>