Polygamma Function - Series Representation

Series Representation

The polygamma function has the series representation

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

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

One more series may be permitted for the polygamma functions. As given by Schlömilch,

. This is a result of the Weierstrass factorization theorem.

Thus, the gamma function may now be defined as:

Now, the natural logarithm of the gamma function is easily representable:

Finally, we arrive at a summation representation for the polygamma function:

Where is the Kronecker delta.

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