# Is there a decomposition for the digamma function as a sum of digamma functions?

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 $$\psi^{[n]}(mz)=\delta_{n,0}\ln m +\frac{1}{m^{n+1}}\sum_{k=0}^{m-1}\psi^{[n]}\left(z+\frac{k}{m}\right)$$