Polygamma Function

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Polygamma Function

Derivative Of The Logarithm of the Gamma Function :

:

\psi^{(m)}(z) = \left(rac{d}{dz}
ight)^m \psi(z) = \left(rac{d}{dz}
ight)^{m+1} \ln\Gamma(z)

Here

:

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

is the Digamma Function and

\Gamma(z)

is the gamma function. The function

\psi^{(1)}(z)

is sometimes called the Trigamma Function .

Integral representation

The polygamma function may be represented as

:

\psi^{(m)}(z)= \int_0^\infty 
rac{t^m e^{-zt}} {1-e^{-t}} dt

which holds for Re ''z'' >0.

Recurrence relation

It has the Recurrence Relation
:

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

Series representation

The polygamma function has the series representation

:

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

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
:

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

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

Taylor's series

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)\; rac {z^k}{k!}

,