Geodesic
On continuants of continued fractions with rational partial quotients
Diskretnaya Matematika, Tome 34 (2022) no. 3, pp. 34-51.

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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.
Keywords: $k$-ary gcd algorithm, continued fraction with rational partial quotients, continuant, triangle of coefficients of Fibonacci polynomials. 
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D. A. Dolgov. On continuants of continued fractions with rational partial quotients. Diskretnaya Matematika, Tome 34 (2022) no. 3, pp. 34-51. http://geodesic.mathdoc.fr/item/DM_2022_34_3_a2/

[1] Sorenson J., The $k$-ary GCD algorithm, Technical Report CS-TR-90-979, University of Wisconsin, Madison, 1990, 18 pp.

[2] Thue A., “Et par antydninger ti1 en taltheoretisk metode”, Kra. Vidensk. Selsk. Forh, 7 (1902), 57–75

[3] Aubry L., “Un théoréme d'arithmétique”, Mathesis, 4:3 (1913)

[4] Vinogradov I. M., “On a general theorem concerning the distribution of the residues and non-residues of powers”, Trans. Amer. Math. Soc., 29 (1927), 209–217 | DOI | MR | Zbl

[5] Sloan N. J. A., Fibonacci polynomials, The On-Line Encyclopedia of Integer Sequences https://oeis.org/wiki/Fibonacci_polynomials

[6] Muir T., A Treatise on the Theory of Determinants, Dover Publ., New York, 1960, 766 pp. | MR

[7] Gelfond A. O., Ischislenie konechnykh raznostei, 2-e izd., GIFML, M., 1959, 400 pp. | MR

[8] Lando S. K., Lektsii o proizvodyaschikh funktsiyakh, MTsNMO, M., 2007, 144 pp.