Skip to main content
SpringerLink
Log in

The Hermite–Pade Approximations of Generalized Hypergeometric Series in Two Variables

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We give examples of well-posed problems of joint Hermite–Pade approximations of series in two variables. We find Rodrigues formulas and integral representations for solutions. We also study the limit distribution of zeros of the corresponding polynomials. Constructions are based, on the one hand, on the classical Appel polynomials orthogonal in a triangle and, on the other hand, on various ways of proving Apery's theorem about irrationality of the number ζ(3).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Suetin P. K., Orthogonal Polynomials in Two Variables [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  2. Bateman H. and ErdÉlyi A., Higher Transcendental Functions. Vol. 2 [Russian translation], Nauka, Moscow (1966).

    Google Scholar 

  3. Appel P. and Kampe de Feriet J., Fonctions hypergeometriques et hyperspheriques, polynomes d'Hermite, Gauthier-Villars, Paris (1926).

    Google Scholar 

  4. Baker G. A. and Graves-Morris P., PadÉ Approximations [Russian translation], Mir, Moscow (1986).

    Google Scholar 

  5. Nikishin E. M. and Sorokin V. N., Rational Approximations and Orthogonality [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  6. Hermite Ch., “Sur la fonction exponentielle,” C. R. Acad. Sci. Paris, 77, 18–24, 74–79, 226–233, 285–293 (1873).

    Google Scholar 

  7. Marcellan F. and Rocha I. A., “On semiclassical linear functionals: integral representations,” J. Comput. Appl. Math., 57, 239–249 (1995).

    Google Scholar 

  8. Beukers F., “PadÉ approximations in number theory,” Lecture Notes in Math., 888, 90–99 (1981).

    Google Scholar 

  9. Apery R., “Irrationalite de ζ(2) et ζ(3). Journees arithmetiques de Luminy,” Asterisque, 61, 11–13 (1979).

    Google Scholar 

  10. Kalyagin V. A., “On one class of polynomials determined by two relations of orthogonality,” Mat. Sb., 110, 609–627 (1979).

    Google Scholar 

  11. Gonchar A. A. and Rakhmanov E. A., “Equilibrium measure and the distribution of zeros of extremal polynomials,” Mat. Sb., 125, No. 1, 117–127 (1984).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, Russia

    V. N. Sorokin

Authors
  1. V. N. Sorokin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sorokin, V.N. The Hermite–Pade Approximations of Generalized Hypergeometric Series in Two Variables. Siberian Mathematical Journal 43, 719–730 (2002). https://doi.org/10.1023/A:1016336605594

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1016336605594

Search

Navigation