For questions about, or related to the polygamma function.
The polygamma function of order $m$ is a meromorphic function on the complex plane defined to be the $(m + 1)$-th derivative of the logarithm of the gamma function; that is,
$$\psi^{(m)}(z) = \frac{d^{m + 1}}{dz^{m + 1}} \ln \Gamma(z)$$
Alternatively, this function can be denoted as $\psi_m$. These functions are holomorphic except at the non-positive integers, where they each have a pole of order $m + 1$.
When $m = 0$, $\psi_0$ is frequently called the digamma function, and $\psi_1$ is called the trigamma function.
These functions satisfy a recurrence relation
$$\psi_n(z + 1) = \psi_n(z) + (-1)^n n! z^{-n - 1}$$
and a reflection formula
$$\psi_n(1 - z) + (-1)^{n + 1} \psi_n(z) = (-1)^n \pi \frac{d^n}{dz^n} \cot(\pi z)$$ For more, see polygamma function on Wolfram MathWorld.
