On multiple orthogonal polynomials for three Meixner measures



Multiple orthogonal polynomials for three discrete Meixner measures with identical exponential decay at infinity are studied. These polynomials are the denominators of the type II Hermite–Padé approximants to some hypergeometric functions. The limit distribution of zeros of such polynomials scaled in a certain way is described in terms of equilibrium logarithmic potentials and in terms of algebraic curves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Arvesú, J. Coussement, and W. Van Assche, “Some discrete multiple orthogonal polynomials,” J. Comput. Appl. Math. 153, 19–45 (2003).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2, Bateman Manuscript Project.MATHGoogle Scholar
  3. 3.
    A. A. Gonchar and E. A. Rakhmanov, “On convergence of simultaneous Padé approximants for systems of functions of Markov type,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 157, 31–48 (1981) [Proc. Steklov Inst. Math. 157, 31–50 (1983)].MATHGoogle Scholar
  4. 4.
    A. A. Gonchar and E. A. Rakhmanov, “Equilibrium measure and the distribution of zeros of extremal polynomials,” Mat. Sb. 125 (1), 117–127 (1984) [Math. USSR, Sb. 53, 119–130 (1986)].MathSciNetMATHGoogle Scholar
  5. 5.
    A. A. Gonchar and E. A. Rakhmanov, “On the equilibrium problem for vector potentials,” Usp. Mat. Nauk 40 (4), 155–156 (1985) [Russ. Math. Surv. 40 (4), 183–184 (1985)].MathSciNetMATHGoogle Scholar
  6. 6.
    A. A. Gonchar, E. A. Rakhmanov, and V. N. Sorokin, “Hermite–Padé approximants for systems of Markov-type functions,” Mat. Sb. 188 (5), 33–58 (1997) [Sb. Math. 188, 671–696 (1997)].MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    N. S. Landkof, Foundations of Modern Potential Theory (Nauka, Moscow, 1966; Springer, Berlin, 1972).CrossRefMATHGoogle Scholar
  8. 8.
    A. Martínez-Finkelshtein, E. A. Rakhmanov, and S. P. Suetin, “Variation of the equilibrium energy and the S-property of stationary compact sets,” Mat. Sb. 202 (12), 113–136 (2011) [Sb. Math. 202, 1831–1852 (2011)].MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    E. M. Nikishin, “On simultaneous Padé approximants,” Mat. Sb. 113 (4), 499–519 (1980) [Math. USSR, Sb. 41 (4), 409–425 (1982)].MathSciNetMATHGoogle Scholar
  10. 10.
    E. M. Nikishin, “The asymptotic behavior of linear forms for joint Padé approximations,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 33–41 (1986) [Sov. Math. 30 (2), 43–52 (1986)].MathSciNetMATHGoogle Scholar
  11. 11.
    E. M. Nikishin and V. N. Sorokin, Rational Approximations and Orthogonality (Nauka, Moscow, 1988; Am. Math. Soc., Providence, RI, 1991).MATHGoogle Scholar
  12. 12.
    E. A. Rakhmanov, “On asymptotic properties of polynomials orthogonal on the real axis,” Mat. Sb. 119 (2), 163–203 (1982) [Math. USSR, Sb. 47 (1), 155–19 (1984)].MathSciNetMATHGoogle Scholar
  13. 13.
    E. A. Rakhmanov, “Equilibrium measure and the distribution of zeros of the extremal polynomials of a discrete variable,” Mat. Sb. 187 (8), 109–124 (1996) [Sb. Math. 187, 1213–1228 (1996)].MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    V. N. Sorokin, “On multiple orthogonal polynomials for discrete Meixner measures,” Mat. Sb. 201 (10), 137–160 (2010) [Sb. Math. 201, 1539–1561 (2010)].CrossRefMATHGoogle Scholar
  15. 15.
    V. N. Sorokin, “On asymptotic regimes for the multiple Meixner polynomials,” Preprint No. 46 (Keldysh Inst. Appl. Math., Moscow, 2016).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia

Personalised recommendations