Let $\psi(x)$ denote the digamma function $$ \psi(x)=\Gamma(x)\frac{\partial}{\partial x} \Gamma(x). $$ Consider $x=x_1 +x_2+\dots +x_m$, where $x_j>0$, for $j=1, \ldots,m$. Is there any formula to decompose $\psi(x)$ in terms of $\psi(x_1),\ldots,\psi(x_m)$?
I know that in the very special case of $m=2$ and $x_1=x_2=x/2$, with $x>0$, Legendre duplication formula allows to claim $$ \psi(x_1+x_2)=\log 2 +\frac{1}{2}\left( \psi(x_1) + \psi(x_2+1/2) \right) $$ and I was wondering whether something more general than that is known in the literature.
I'm pretty sure the answer is no. In the case that $x_1=...=x_m=z$ however there is the nice formula $$\psi^{[n]}(mz)=\delta_{n,0}\ln m +\frac{1}{m^{n+1}}\sum_{k=0}^{m-1}\psi^{[n]}\left(z+\frac{k}{m}\right)$$