Polygamma function

The polygamma function is a special function that is defined as the \( n \)th derivative of the logarithm of the gamma function, denoted as \( \psi^{(n)}(x) \). Specifically, it is defined as: \[ \psi^{(n)}(x) = \frac{d^n}{dx^n} \ln(\Gamma(x)) \] where \( \Gamma(x) \) is the gamma function.