While studying Gamma function and related functions I noticed that its logarithmic derivative (the so-called Digamma function) is studied more than its "normal" derivative but on the other hand I can't find anything on the logarithmic derivative of the Digamma function but only its usual derivatives (the so-called Polygamma functions).
So my question is: have already been studied the logarithmic derivatives of the Polygamma functions and they are not interesting (and maybe why they are not?) or none simply ever cared about ?
The $\Gamma$ function is defined as the analytic continuation (through $\Gamma(z+1)=z\,\Gamma(z)$) of $$ \Gamma(\alpha)=\int_{0}^{+\infty} t^{\alpha-1}e^{-t}\,dt,\qquad\text{Re}(\alpha)>0 $$ so it is a meromorphic function over $\mathbb{C}$ with simple poles at $z=0,-1,-2,\ldots$.
By the Mittag-Leffler theorem or through other ways (for instance, the dominated convergence theorem that leads to the Euler product) we have that $\Gamma(z+1)$ can be represented by an infinite product: $$ \Gamma(z+1) = e^{-\gamma z}\prod_{n\geq 1}\left(1+\frac{z}{n}\right)^{-1}e^{\frac{z}{n}}\tag{1}$$ that is the Weierstrass product. Hence $\log\Gamma $ can be represented by a series, as well as: $$ \psi(z+1) = \frac{d}{dz}\log\Gamma(z+1) = -\gamma+\sum_{n\geq 1}\left(\frac{1}{n}-\frac{1}{n+z}\right).\tag{2}$$ We also have an integral representation: $$ \psi(z+1)=-\gamma+\int_{0}^{1}\frac{1-t^z}{1-t}\,dt,\qquad \text{Re}(z)>-1\tag{3}$$ but the functional equation $\psi(z+1)=\frac{1}{z}+\psi(z)$ does not lead to a practical representation of the digamma function as a product. That is the reason for which we never met the logarithmic derivative of the digamma function: despite it just is $\frac{\psi'}{\psi}$, $\frac{\psi'}{\psi}$ does not satisfy a nice functional equation.