$q$-Stern Polynomials as Numerators of Continued Fractions

We present a $q$-analogue for the fact that the $n$th Stern polynomial $B_n(t)$ in the sense of Klav\v zar, Milutinović and Petr [Adv. Appl. Math. 39 (2007)] is the numerator of a continued fraction of $n$ terms. Moreover, we give a combinatorial interpretation for our $q$-analogue.