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.

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