# 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:

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

Here

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

is the digamma function and [itex]\Gamma(z)[itex] is the gamma function.

It has the recurrence relation

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

It is related to the Hurwitz zeta function

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

The Taylor series at z=1 is

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

(-1)^{m+k+1} (m+k)!\; \zeta (m+k+1)\; \frac {z^k}{k!}[itex], which converges for |z|<1. Here, [itex]\zeta(n)[itex] is the Riemann zeta function.

## References

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
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