Q-Polygamma function
From specialfunctionswiki
The $q$-polygamma functions of order $m$, $\psi_q^{(m)}$, are analogues of the polygamma function defined by $$\psi_q^{(m)}(z)=\dfrac{\partial^m}{\partial z^m} \psi_q(z),$$ where $\psi_q(z) = \dfrac{1}{\Gamma_q(z)} \dfrac{\partial}{\partial z} \Gamma_q(z).$ Here the function $\Gamma_q$ is the $q$-Gamma function.
