I am a fan of the Matrix exponential $\exp(X)$, defined for any complex matrix $X$ by \begin{equation*} \exp(X) := \sum_{k \ge 0} \frac{X^k}{k!}. \end{equation*}

I have a fleeting acquaintance with q-analog's (essentially, I know that they exist, but have almost no idea what use they serve, which is part of the reason why I am asking this question).

Thus, my question is

Has the following q-analog of the matrix exponential \begin{equation*} \exp_q(X) := \sum_{k \ge 0} \frac{X^k}{[k]_q!}, \end{equation*} been studied previously? If so, in what context?

*PS:* More generally, the above question can be rephrased in terms of q-analogs of functions of matrices (which includes scalar, vector, and matrix valued functions). But I wanted to limit my focus to a concrete case of special interest to me.