Advertisement

Mathematical Notes

, Volume 78, Issue 5–6, pp 827–840 | Cite as

A Generalization of Pincherle's Theorem to k-Term Recursion Relations

  • V. I. Parusnikov
Article

Abstract

In 1894, Pincherle proved a theorem relating the existence of a minimal solution of three-term recursion relations to the convergence of a continued fraction. The present paper deals with solutions of an infinite system
$$q_n = \sum\limits_{j - 1}^{k - 1} {_{Pk - j,n} } q_{n - j} ,\quad p_{1,n} \ne 0,\quad n = 0,1, \ldots ,$$
of k-term recursion relations with coefficients in a field F. We study the connection between such relations and multidimensional ((k − 2)-dimensional) continued fractions. A multidimensional analog of Pincherle's theorem is established.

Key words

Pincherle theorem recursion relation multidimensional continued fraction 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    W. Gautschi, “Computational aspects of three-term recurrence relations,” SIAM Review, 9 (1967), 24–82.CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    S. Pincherle, “Delle funzioni ipergeometriche e di varie questioni ad esse attinenti,” Giorn. Mat. Battaglini, 32 (1894), 209–291.MATHGoogle Scholar
  3. 3.
    W. B. Jones and W. J. Thron, Continued Fractions. Analytic Theory and Applications, Addison-Wesley, Reading, Mass., 1980. Russian translation: Mir, Moscow, 1985.Google Scholar
  4. 4.
    V. I. Parusnikov, A Generalization of Pincherle's Theorem for k-Term Recursion Relations [in Russian], Preprint of Inst. of Appl. Math., Russ. Acad. Sci., Moscow, 2000, no. 87.Google Scholar
  5. 5.
    C. G. J. Jacobi, “Allgemeine Theorie der kettenbruchahnlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden gebildet wird,” J. Reine Angew. Math., 69 (1868), 29–64.MATHGoogle Scholar
  6. 6.
    L. Euler, “De relatione inter ternas pluresve quantitates instituenda,” Petersburger Akademie Notiz, Exhib. August 14, 1775.Google Scholar
  7. 7.
    O. Perron, “Grundlagen fur eine Theorie des Jakobischen Kettenbruchalgorithmus,” Math. Ann., 64 (1907), no. 6, 1–76.MATHMathSciNetGoogle Scholar
  8. 8.
    R. E. A. C. Paley and H. D. Ursell, “Continued fractions in several dimensions,” Proc. Cambridge Phil. Soc., 26 (1930), no. 2, 127–144.Google Scholar
  9. 9.
    V. I. Parusnikov, “The Jacobi-Perron algorithm and simultaneous approximation of functions,” Mat. Sb. [Math. USSR-Sb.], 114(156) (1981), no. 2, 322–333.MATHMathSciNetGoogle Scholar
  10. 10.
    Bruijn M. G. de, “The interruption phenomenon for generalized continued fractions,” Bull. Austral. Math. Soc., 19 (1978), no. 2, 245–272.MathSciNetGoogle Scholar
  11. 11.
    A. O. Gel'fond, The Calculus of Finite Differences [in Russian], Nauka, Moscow, 1967.Google Scholar
  12. 12.
    V. I. Parusnikov, Limit-Periodic Multidimensional Continued Fractions [in Russian], Preprint of Inst. of Appl. Math., Acad. Sci. USSR, Moscow, 1983, no. 62.Google Scholar
  13. 13.
    V. I. Parusnikov, “On the convergence of the multidimensional limit-periodic continued fractions,” Lect. Notes in Math. (1985), no. 1237, 217–227.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • V. I. Parusnikov
    • 1
  1. 1.Institute of Applied MathematicsRussian Academy of SciencesRussia

Personalised recommendations