A certain class of approximations for the $q$-digamma function

Abstract

In this paper, we derive a class of approximations of the $q$-digamma function $\psi _q(x)$. The infinite family \[ I_a(x;q)=\log [x+a]_q+\frac {q^x\log q}{1-q^x}-\bigg (\frac 12-a\bigg )H(q-1)\log q, \] $a\in [0,1]$; $q>0$, can be used as approximating functions for $\psi _q(x)$, where $[x]_q=(1-q^x)/(1-q)$ and $H(\cdot )$ is the Heaviside step function. We show that, for all $a\in [0,1]$, $I_a$ is asymptotically equivalent to $\psi _q(x)$ for $q>0$ and is a good pointwise approximation.