I would like to know if there's a quicker way to verify:
$$\partial_z^{n-1}\psi(tz)t^n = \partial_z^n\ln[\Gamma(tz)], \,\,\\ n \in \mathbb{N}^+, t \in \mathbb{C}_+\tag{1}\label{1}$$
That's true for $t=1$. But what about a complex $t$?
For complex $t$, $(1)$ holds for $n=1,2,3$. These computations were done by hand. I searched any information that would help me on DLMF (https://dlmf.nist.gov/5.15), but nothing relevant was found.
Thanks
By the definition, $$\psi(x) = \dfrac{d}{dx} \ln \Gamma(x).$$
Therefore, $$\dfrac{\partial^{n}}{\partial z^n}\ln\Gamma(t z) = \dfrac{\partial^{n-1}}{\partial z^{n-1}}\left(\dfrac{\partial}{\partial z}\ln\Gamma(t z)\right) = \dfrac{\partial^{n-1}}{\partial z^{n-1}}\Big(t\, \psi(t z)\Big)= t^{\color{brown}{\mathbf{1}}}\dfrac{\partial^{n-1}}{\partial z^{n-1}} \psi(t z)$$
(see also WA calculations for $n=3$).