Computational Methods and Function Theory

, Volume 16, Issue 2, pp 167–185 | Cite as

Asymptotic Distribution of Zeros of a Certain Class of Hypergeometric Polynomials

  • Addisalem Abathun
  • Rikard Bøgvad


We study the asymptotic behavior of the zeros of a family of a certain class of hypergeometric polynomials \(\small {}_{A}\text {F}_{B}\left[ \begin{array}{c} -n,a_2,\ldots ,a_A \\ b_1,b_2,\ldots ,b_B \end{array} ; \begin{array}{cc} z \end{array}\right] \), using the associated hypergeometric differential equation, as the parameters go to infinity. The curve configuration on which the zeros cluster is characterized as level curves associated with integrals on an algebraic curve. The algebraic curve is the hypergeometrc differential equation, using a similar approach to the method used in Borcea et al. (Publ Res Inst Math Sci 45(2):525–568, 2009). In a specific degenerate case, we make a conjecture that generalizes work in Boggs and Duren (Comput Methods Funct Theory 1(1):275–287, 2001), Driver and Duren (Algorithms 21(1–4):147–156, 1999), and Duren and Guillou (J Approx Theory 111(2):329–343, 2001), and present experimental evidence to substantiate it.


Hypergeometric polynomials Cauchy transform Asymptotic zero measures Hypergeometric differential equation 

Mathematics Subject Classification

33C05 33C20 31A35 34E05 


  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions, Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  2. 2.
    Bergkvist, T., Hans, R.: On polynomial eigenfunctions for a class of differential operators. Math. Res. Lett. 9(2–3), 153–171 (2002)Google Scholar
  3. 3.
    Björk, J.-E., Borcea, J., Bøgvad, R.: Subharmonic configuration and alge-braic cauchy transform of probability measure. In: Bränden, P., Passare, M., Putinar, M. (eds.) Notions of Positivity and the Geometry of Polynomials. Trends in Mathematics, Birkhauser, Basel (2011)Google Scholar
  4. 4.
    Boggs, K., Duren, P.: Zeros of hypergeometric functions. Comput. Methods Funct. Theory 1(1), 275–287 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borcea, J., Bøgvad, R.: Piecewise harmonic subharmonic functions and positive Cauchy transforms. Pac. J. Math. 240(2), 231–265 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borcea, J., Bøgvad, R., Shapiro, B.: Homogenized spectral problems for exactly solvable operators: asymptotics of polynomial eigenfunctions. Publ. Res. Inst. Math. Sci. 45(2), 525–568 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Borcea, J., Bøgvad, R., Shapiro, B.: Corrigendum: homogenized spectral problems for exactly solvable operators: asymptotics of polynomial eigenfunctions. Publ. Res. Inst. Math. Sci. 48(1), 229–233 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Driver, K., Duren, P.: Asymptotic zero distribution of hypergeometric polynomials. Numer. Algorithms 21(1–4), 147–156 (1999). Numerical methods for partial differential equations (Marrakech, 1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Duren, P.L., Guillou, B.J.: Asymptotic properties of zeros of hypergeometric polynomials. J. Approx. Theory 111(2), 329–343 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kuijlaars, A.B.J., Martínez-Finkelshtein, A.: Strong asymptotics for Jacobi polynomials with varying nonstandard parameters. J. Anal. Math. 94, 195–234 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Martínez-Finkelshtein, A., Martínez-González, P., Orive, R.: Zeros of Jacobi polynomials with varying non-classical parameters, Special functions (Hong Kong, 1999) (2000) pp. 98–113Google Scholar
  12. 12.
    Martínez-Finkelshtein, A., Orive, R.: Riemann-Hilbert analysis of Jacobi polynomials orthogonal on a single contour. J. Approx. Theory 134(2), 137–170 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Slater, L.J.: Generalized hypergeometric functions. Cambridge University Press, Cambridge (1966)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsAddis Ababa UniversityAddis AbabaEthiopia
  2. 2.Department of MathematicsStockholm UniversityStockholmSweden

Personalised recommendations