On continuants of continued fractions with rational partial quotients

Continued fractions with rational partial quotients with right shift arise in the process of applying Sorenson's right-shift $k$-ary gcd algorithm to the ratio of natural numbers $a$, $b$. Using this algorithm makes it possible to obtain different types of such fractions. This functions are associated with special forms of continuants, that is, polynomials that can be used to express the numerators and denominators of the convergents. In this paper we introduce such fractions and continuants, we also investigate properties (in particular, the asymptotic behavior) of the extremal values of the continuants under constraints imposed on the variables involved in the right-shift $k$-ary gcd algorithm of Sorenson. We also introduce a construction similar to the triangle of coefficients of Fibonacci polynomials.