On continuants of continued fractions with rational partial quotients
-
Dmitry A. Dolgov
Abstract
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.
Originally published in Diskretnaya Matematika (2022) 34, №3, 34–51 (in Russian).
References
[1] Sorenson J., “The k-ary GCD algorithm”, Technical Report CS-TR-90-979, University of Wisconsin, Madison, 1990, 18 pp.Search in Google Scholar
[2] Thue A., “Et par antydninger ti1 en taltheoretisk metode”, Kra. Vidensk. Selsk. Forh, 7 (1902), 57–75.Search in Google Scholar
[3] Aubry L., “Un théoréme d’arithmétique”, Mathesis, 4:3 (1913)Search in Google Scholar
[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), 209–217.10.1090/S0002-9947-1927-1501384-3Search in Google Scholar
[5] Sloan N. J. A., “Fibonacci polynomials”, The On-Line Encyclopedia of Integer Sequences, https://oeis.org/wiki/Fibonacci_polynomials.Search in Google Scholar
[6] Muir T., A Treatise on the Theory of Determinants, Dover Publ., New York, 1960, 766 pp.Search in Google Scholar
[7] Gel’fond A. O., Calculus of Finite Differences, Hindustan Publ. Comp., 1971, 451 pp.Search in Google Scholar
[8] Lando S. K., Lectures on Generating Functions, M.: MCCME, 2007 (in Russian), 144 pp.Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On continuants of continued fractions with rational partial quotients
- On linear equivalence of piecewise-linear permutations of the field 𝔽2n
- On the prospective minimum of the random walk conditioned to stay nonnegative
- Limiting behavior of percolation cluster in a multilayered random environment with breakdown
Articles in the same Issue
- Frontmatter
- On continuants of continued fractions with rational partial quotients
- On linear equivalence of piecewise-linear permutations of the field 𝔽2n
- On the prospective minimum of the random walk conditioned to stay nonnegative
- Limiting behavior of percolation cluster in a multilayered random environment with breakdown
