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Ukrainian Mathematical Journal

, Volume 62, Issue 3, pp 358–372 | Cite as

On Hankel determinants of functions given by their expansions in P-fractions

  • V. I. Buslaev
Article

We obtain explicit formulas that express the Hankel determinants of functions given by their expansions in continued P-fractions in terms of the parameters of the fraction. As a corollary, we obtain a lower bound for the capacity of the set of singular points of these functions, an analog of the van Vleck theorem for P-fractions with limit-periodic coefficients, another proof of the Gonchar theorem on the Leighton conjecture, and an upper bound for the radius of the disk of meromorphy of a function given by a C-fraction.

Keywords

Singular Point Meromorphic Function Continue Fraction Continue Fraction Expansion Nondecreasing Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    W. B. Jones and W. J. Thron, Continued Fractions. Analytic Theory and Applications [Russian translation], Mir, Moscow (1985).Google Scholar
  2. 2.
    G. Pólya, “Über gewisse notwendige Determinantkriterien für die Forsetzbarkeit einer Potenzreihe,” Math. Ann., 99, 687–706 (1928).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. A. Gonchar, “On singular points of meromorphic functions defined by their expansion in a C-fraction,” Mat. Sb., 197, No. 10, 3–14 (2006).MathSciNetGoogle Scholar
  4. 4.
    H. S. Wall, Analytic Theory of Continued Fractions, van Nostrand, New York (1948).Google Scholar
  5. 5.
    J. Worpitsky, “Untersuchungen über die Entwickelung der monodromen und monogenen Funktionen durch Kettenbruche,” in: Friedrichs-Gymnasium rund Realschule Jahresbericht, Berlin (1865), pp. 3 –39.Google Scholar
  6. 6.
    W. T. Scott and H. S. Wall, “Continued fraction expansions for arbitrary power series,” Ann. Math., 41, No. 2, 328–349 (1940).CrossRefMathSciNetGoogle Scholar
  7. 7.
    W. J. Thron, “Twin convergence regions for continued fractions b 0 + K(1, b n), II,” Amer. J. Math., 71, 112–120 (1949).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    G. M. Goluzin, Geometric Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1966).Google Scholar
  9. 9.
    N. P. Callas and W. J. Thron, “Singularities of meromorphic functions represented by regular C-fractions,” Kgl. Norske Vid. Selsk. Skr. (Trondheim), No. 6, 11 (1967).Google Scholar
  10. 10.
    V. I. Buslaev, “On the convergence of the Rogers–Ramanujan continued fraction,” Mat. Sb., 194, No. 6, 43–66 (2003).MathSciNetGoogle Scholar
  11. 11.
    E. V. van Vleck, “On the convergence of algebraic continued fractions whose coefficients have limiting values,” Trans. Amer. Math. Soc., 5, No. 5, 253–262 (1904).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    V. I. Buslaev, “On the van Vleck theorem for regular C-fractions with limit-periodic coefficients,” Izv. Ros. Akad. Nauk, Ser. Mat., 65, No. 4, 35–48 (2001).MathSciNetGoogle Scholar
  13. 13.
    H. Stahl, “Orthogonal polynomials with complex-valued weight functions. I, II,” Constr. Approxim., 2, 225–251 (1986).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  • V. I. Buslaev
    • 1
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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