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

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

is the digamma function and $Γ(z)$ is the gamma function.

It has the recurrence relation

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

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

The Taylor series at z=1 is

$\psi^{(m)}(z+1)= \sum_{k=0}^\infty (-1)^{m+k+1} (m+k)!\; \zeta (m+k+1)\; \frac {z^k}{k!}$ ,

which converges for |z|<1. Here, $ζ(n)$ 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