Polygamma Function - Taylor Series

Taylor Series

The Taylor series at z = 1 is

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

and

which converges for |z| < 1. Here, ΞΆ is the Riemann zeta function. This series is easily derived from the corresponding Taylor series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series.

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