How does one prove the polygamma reflection formula:
$$\psi^{(n)}(1-z)+(-1)^{n+1}\psi^{(n)}(z)=(-1)^n \pi \frac{d^n}{d z^n} \cot \pi z $$
Do we have to invoke the power of contour integration and kernels? I searched the web but the proof was to be found nowhere.
You just need to prove the reflection formula: $$ \psi(1-z)-\psi(z)=\pi\cot(\pi z)\tag{1}$$ then differentiate it multiple times. In order to prove $(1)$, let's start from the Weierstrass product for the $\Gamma$ function: $$\Gamma(t+1) = e^{-\gamma t}\prod_{n=1}^{+\infty}\left(1+\frac{t}{n}\right)^{-1}e^{\frac{t}{n}}\tag{2}$$ leading to: $$ \Gamma(z)\,\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}\tag{3} $$ (the reflection formula for the $\Gamma$ function), then consider the logarithmic derivative of $(3)$: $$\psi(z)-\psi(1-z) = \frac{d}{dz}\,\log\sin(\pi z)\tag{4} $$ and $(1)$ is proved.