On zeros of numerator and denominator polynomials of Thiele’s continued fraction

Abstract

UDC 517.518:519.652

We prove that the polynomials of canonical numerators and denominators of the interpolation and approximation convergents of Thiele’s continued fractions have no common zeros. It is established that the convergents of Thiele’s continued fraction form a staircase sequence of normal Pad´e approximants. The region of zeros of the denominator polynomial of the convergent of Thiele’s continued fraction is also determined.

References

J. M. Hoëné – Wro´nski, Introduction à la philosophie des mathématiques et Technie de l’Algorithmique. (French), Courcier, Paris (1811)

J. M. Hoëné –Wro´nski, Philosophie de la Technie Algorithmique: Loi Suprˆeme et universelle des Math´ematiques, de L’imprimerie de P. Didot L’Aine, Paris (1815 – 1817).

T. N. Thiele, Interpolationsprechnung, Commisission von B.G. Teubner, Leipzig (1909).

N. E. N¨orlund, Vorlesungen ¨uber Differenzenrechnung, Springer, Berlin (1924). DOI: https://doi.org/10.1007/978-3-642-50824-0

C. Brezinski, History of continued fractions and pade approximations, vol. 12, Springer Sci. & Business Media (2012), https://doi.org/10.1007/978-3-642-58169-4 DOI: https://doi.org/10.1007/978-3-642-58169-4

N. S. Bahvalov, N. P. Zhitkov, G. M. Kobel'kov, Chislennye metody, Nauka, Moskva (1987).

F. B. Hildebrand, Introduction to numerical analysis. 2-nd ed., Dover Publ., Inc., New York (1987).

Sh. E. Mikeladze, Chislennye metody matematicheskogo analiza, Gostekhteorizdat, Moskva(1953).

U. Dzhouns, V.Tron, Непрерывные дроби. (Russian) [[Continued fractions]] Аналитическая теория и приложения. [Analytic theory and applications], ``Mir, Moscow (1985).

A. Cuyt, V. Brevik Petersen, B. Verdonk, H. Waadeland, W. B. Jones, Handbooks of Continued Fractions for Special Functions, Springer, Berlin etc. (2008).

V. Ya. Skorobogatʹko, Теория ветвящихся цепных дробей и ее применение в вычислительной математике. (Russian) [[The theory of branching continued fractions and its application in numerical mathematics]], Nauka, Moscow (1983).

J. Tan, Theory of continued fractions and its applications, Sci. Publ., Beijing (2007).

P. Henrici, P. Pfluger, Truncation error estimates for Stieltjes fractions, Numer. Math., 9, 120 – 138 (1966), https://doi.org/10.1007/BF02166031 DOI: https://doi.org/10.1007/BF02166031

G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary school and for colleges, vol. 2, A. & C. Black (1889).

R. Vein, P. Dale, Determinants and their application in mathematical physics, Springer Sci. & Business Media (2006).

M. M. Pagirya, Vikoristannya kontinuanti dlya ocinki zalishkovogo chlena interpolyacijnogo lancyugovogo drobu Tile, Ukr. mat. vis.,16, № 4, 588 – 603 (2019).

M. M. Pagirya, Nablizhennya funkcij lancyugovimi drobami, Grazhda, Uzhgorod(2016).

O. Perron, Die Lehre von den Kettenbr¨uchen, Bd 1, Teubner, Stuttgart (1954).

A. N. Khovanskii, The application of continued fractions and their generalizations to problems in approximation theory, P. Noordhoff Groningen (1963).

P. Henrici, applied and computational complex analysis, vol. 2, special functions – integral transforms – asymptotics – continued fractions, New York etc., Wiley – Intersci. Publ., (1977).

Dzh. Bejker (mol.), P. Grejvs –Morris, Approksimacii Pade, Mir, Moskva(1986).

M. M. Pahirya, R. A. Katsala, Equivalence of two methods for construction of regular continued $C$-fractions, Ukr. Math. J., 61, № 7, 1192 – 1198 (2009), https://doi.org/10.1007/s11253-009-0268-z DOI: https://doi.org/10.1007/s11253-009-0268-z

D. S. Mitrinovi´c, Analytic Inequalities, Berlin etc. Springer (1970). DOI: https://doi.org/10.1007/978-3-642-99970-3_3

M. M. Pahirya, Evaluation of the remainder term for the Thiele interpolation continued fraction, Ukr. Math. J., 60, № 11, 1813 – 1822 (2008), https://doi.org/10.1007/s11253-009-0171-7 DOI: https://doi.org/10.1007/s11253-009-0171-7

M. M. Pahirya, Estimation of the Remainder for the Interpolation Continued C-Fraction, Ukr. Math. J., 66, № 6, 905 – 915 (2014), https://doi.org/10.1007/s11253-014-0980-1 DOI: https://doi.org/10.1007/s11253-014-0980-1

Published
24.01.2022
How to Cite
Pahirya, M. M. “On Zeros of Numerator and Denominator Polynomials of Thiele’s Continued Fraction”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 1, Jan. 2022, pp. 113 -21, doi:10.37863/umzh.v74i1.6133.
Section
Research articles