Continued fractions and polynomials related to hyperbinary representations

Schinzel recently showed that the $n$th Stern polynomial of Klavžar et al. is the numerator of a certain finite continued fraction. This was subsequently extended by Mansour to $q$-Stern polynomials. We extend these results further to a $2$-parameter bivariate analogue of the sequence of Stern polynomials which arise naturally in the characterization of hyperbinary representations of a given integer. In the process we define a class of companion polynomials with which we can determine the denominators of the continued fractions in question.