Polygamma function

From Academic Kids

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) + (-1)^m\; m!\; z^{-(m+1)}<math>

It is related to the Hurwitz zeta function

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

The Taylor series at z=1 is

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

(-1)^{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.

See also

References

it:Funzione poligamma

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