Polygamma function

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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}{dz}\right)^{m+1} \ln\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. The function <math>\psi^{(1)}(z)</math> is sometimes called the trigamma function.

1 Integral representation
2 Recurrence relation
3 Series representation
4 Taylor's series
5 References

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Integral representation

The polygamma function may be represented as

<math>\psi^{(m)}(z)= \int_0^\infty

\frac{t^m e^{-zt}} {1-e^{-t}} dt</math>

which holds for Re z >0.

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Recurrence relation

It has the recurrence relation

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Series representation

The polygamma function has the series representation

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

\frac{1}{(z+k)^{m+1}}</math>

which holds for m > 0 and any complex z not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as

alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.

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Taylor's series

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. This series is easily derived from the corresponding Taylor's series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series.

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