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In mathematics, the polygamma function of order m is defined as the m+1 'th
derivative of the logarithm of the gamma function:
- <math>\psi^{(m)}(z) = \left(\frac{d}{dz}\right)^m \psi(z) = \left(\frac{d}{dx}\right)^{m+1} \log\Gamma(z)<math>
Here
- <math>\psi(z) =\psi^0(z) = \frac{\Gamma'(z)}{\Gamma(z)}<math>
is the digamma function and <math>\Gamma(z)<math> is the gamma function.
It has the recurrence relation
- <math>\psi^{(m)}(z+1)= \psi^{(m)}(z) + (-)^m\; m!\; z^{-(m+1)}<math>
It is related to the Hurwitz zeta function
- <math>\psi^{(m)}(z) = (-)^{m+1}\; m!\; \zeta (m+1,z)<math>
The Taylor series at z=1 is
- <math>\psi^{(m)}(z+1)= \sum_{k=0}^\infty
(-)^{m+k+1} (m+k)!\; \zeta (m+k+1)\; \frac {z^k}{k!}<math>,
which converges for |z|<1. Here, <math>\zeta(n)<math> is the Riemann zeta function.
References
- Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . See section 6.4.
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