Generalized the q-Digamma and the q-Polygamma Functions via Neutrices

Abstract

The $q$-digamma function $\psi_q(x)$ and the $q$-polygamma functions $\psi^{(r)}_q(x),~r\in\N$ are defined for all $x>0$ and $0<q<1$. In this paper, the neutrices and the neutrix limit are used to redefine the $q$-digamma function $\psi_q(x)$ and the $q$-polygamma functions $\psi^{(r)}_q(x),~r\in\N$ for all $x\in\R$. Also, further results are given.