polygamma function

In mathematics, the polygamma function of order ''m is defined as the m''+1 'th derivative of the logarithm of the gamma function:

\psi^{(m)}(z) = \left(\frac{d}{dz}\right)^m \psi(z) = \left(\frac{d}{dx}\right)^{m+1} \log\Gamma(z)

Here

\psi(z) =\psi^0(z) = \frac{\Gamma'(z)}{\Gamma(z)}

\Gamma(z)

is the gamma function. It has the recurrence relation

\psi^{(m)}(z+1)= \psi^{(m)}(z) + (-)^m\; m!\; z^{-(m+1)}

It is related to the Hurwitz zeta function

\psi^{(m)}(z) = (-)^{m+1}\; m!\; \zeta (m+1,z)

The Taylor series at z=1 is

\psi^{(m)}(z+1)= \sum_{k=0}^\infty

(-)^{m+k+1} (m+k)!\; \zeta (m+k+1)\; \frac {z^k}{k!}, which converges for |z|<1. Here,

\zeta(n)

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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