Skip to main content
SpringerLink
Log in

On the zeros of Meixner polynomials

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We investigate the zeros of a family of hypergeometric polynomials \(M_n(x;\beta ,c)=(\beta )_n\,{}_2F_1(-n,-x;\beta ;1-\frac{1}{c})\), \(n\in \mathbb N ,\) known as Meixner polynomials, that are orthogonal on \((0,\infty )\) with respect to a discrete measure for \(\beta >0\) and \(0<c<1.\) When \(\beta =-N\), \(N\in \mathbb N \) and \(c=\frac{p}{p-1}\), the polynomials \(K_n(x;p,N)=(-N)_n\,{}_2F_1(-n,-x;-N;\frac{1}{p})\), \(n=0,1,\ldots , N\), \(0<p<1\) are referred to as Krawtchouk polynomials. We prove results for the zero location of the orthogonal polynomials \(M_n(x;\beta ,c)\), \(c<0\) and \(n<1-\beta \), the quasi-orthogonal polynomials \(M_n(x;\beta ,c)\), \(-k<\beta <-k+1\), \(k=1,\ldots ,n-1\) and \(0<c<1\) or \(c>1,\) as well as the polynomials \(K_{n}(x;p,N)\) with non-Hermitian orthogonality for \(0<p<1\) and \(n=N+1,N+2,\ldots \). We also show that the polynomials \(M_n(x;\beta ,c)\), \(\beta \in \mathbb R \) are real-rooted when \(c\rightarrow 0\).

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Álvarez de Morales, M., Pérez, T., Piñar, M., Ronveaux, A.: Non-standard orthogonality for Meixner polynomials. Electron. Trans. Numer. Anal. 9, 1–25 (1999)

  2. Álvarez-Nodarse, R., Dehesa, J.S.: Distributions of zeros of discrete and continuous polynomials from their recurrence relation. Appl. Math. Comput. 128(2–3), 167–190 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Askey, R., Wilson, J.: A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols. SIAM J. Math. Anal. 10, 1008–1016 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Askey, R., et al.: Evaluation of Sylvester type determinants using orthogonal polynomials. In: Begehr, H.G.W. (ed.) Advances in Analysis: Proceedings of 4th International ISAAC Congress, pp. 1–16. World Scientific, Singapore (2005)

    Chapter  Google Scholar 

  5. Atkinson, F.V.: Discrete and Continuous Boundary Problems. Mathematics in Science and Engineering. Academic Press, New York (1964)

    Google Scholar 

  6. Braenden, P.: On operators on polynomials preserving real-rootedness and the Neggers–Stanley conjecture. J. Algebr. Comb. 20, 119–130 (2004)

    Article  MATH  Google Scholar 

  7. Brenti, F.: Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update Jerusalem Combinatorics ’93, Contemporary Mathematics, vol. 178, pp. 71–89. American Mathematical Society, Providence (1994)

  8. Brezinski, C., Driver, K.A., Redivo-Zaglia, M.: Quasi-orthogonality with applications to some families of classical orthogonal polynomials. Appl. Numer. Math. 48, 157–168 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chihara, L., Stanton, D.: Zeros of generalized Krawtchouk polynomials. J. Approx. Theory 60(1), 43–57 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)

    MATH  Google Scholar 

  11. Chihara, T.S.: On quasi-orthogonal polynomials. Proc. Amer. Math. Soc. 8, 765–767 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  12. Costas-Santos, R.S., Sánchez-Lara, J.F.: Extensions of discrete classical orthogonal polynomials beyond the orthogonality. J. Comput. Appl. Math. 225, 440–451 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Costas-Santos, R.S., Sánchez-Lara, J.F.: Orthogonality of \(q\)-polynomials for non-standard parameters. J. Approx. Theory 163(9), 1246–1268 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dragnev, P.D., Saff, E.B.: Constrained energy problems with applications to orthogonal polynomials of a discrete variable. J. Anal. Math. 72, 223–259 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dragnev, P.D., Saff, E.B.: A problem in potential theory and zero asymptotics of Krawtchouk polynomials. J. Approx. Theory 102(1), 120–140 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Driver, K., Jordaan, K.: Convergence of ray sequences of Padé approximants to \(_2F_1(a,1;c;z)\). Quaest. Math. 25, 1–7 (2002)

    Article  MathSciNet  Google Scholar 

  17. Driver, K., Jordaan, K., Martínez-Finkelshtein, A.: Pólya frequency sequences and real zeros of some \(_3F_2\) polynomials. J. Appl. Math. Anal. Appl. 332(2), 1045–1055 (2007)

    Article  MATH  Google Scholar 

  18. Grafova, I.B., Grafov, B.M.: Meixner wavelet transform: a tool for studying stationary discrete-time stochastic processes. Russ. J. Electrochem. 39(2), 130–133 (2003)

    Article  Google Scholar 

  19. Holtz, O., et al.: Evaluation of Sylvester type determinants using block-triangularization. In: Begehr, H.G.W. (ed.) Advances in Analysis: Proceedings of 4th International ISAAC Congress, pp. 395–405. World Scientific, Singapore (2005)

    Chapter  Google Scholar 

  20. Ismail, M.E.H., Simeonov, P.: Strong asymptotics for Krawtchouk polynomials. J. Comput. Appl. Math. 100(2), 121–144 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jin, X., Wong, R.: Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96, 281–300 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jordan, C.: Sur une série de polynomes. Proc. Lond. Math. Soc. 20(2), 297–325 (1920)

    Google Scholar 

  23. Joulak, H.: A contribution to quasi-orthogonal polynomials and associated polynomials. Appl. Numer. Math. 54(1), 65–78 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric orthogonal polynomials and their \(q\)-analogues. Springer, Berlin (2010)

  25. Koornwinder, T.H.: Krawtchouk polynomials, a unification of two different group theoretic interpretations. SIAM J. Math. Anal. 13, 1011–1023 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  26. Krasovsky, I.V.: Asymptotic distribution of zeros of polynomials satisfying difference equations. J. Comput. Appl. Math. 150(1), 57–70 (2003)

    Google Scholar 

  27. Kuijlaars, A.B.J., Martínez-Finkelshtein, A.: Strong asymptotics for Jacobi polynomials with varying nonstandard parameters. J. Anal. Math. 94, 195–234 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Levit, R.J.: The zeros of the Hahn polynomials. SIAM Rev. 9(2), 191–203 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  29. Martínez-Finkelshtein, A., Martínez-González, P., Orive, R.: Zeros of Jacobi polynomials with varying non-classical parameters, Special functions. In: Dunkl, C., et al. (eds.) Proceedings of the International Workshop on Special Functions: Asymptotics, Harmonic Analysis and Mathematical Physics, Hong Kong, China, June 21–25, 1999, pp. 98–113. World Scientific, Singapore (2000)

    Google Scholar 

  30. Porter, M.: On the roots of functions connected by a linear recurrent relation of the second order. Ann. Math. 3(2), 55–70 (1902)

    MATH  Google Scholar 

  31. Rainville, E.D.: Special Functions. The Macmillan Company, New York (1960)

    MATH  Google Scholar 

  32. Sorokin, V.N.: On multiple orthogonal polynomials for discrete Meixner measures. Sb. Math. 201(10), 1539–1561 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Stanley, R.: Log-concave and unimodal sequences in algebra, combinatorics and geometry. Ann. N Y Acad. Sci. 576, 500–534 (1989)

    Article  MathSciNet  Google Scholar 

  34. Sylvester, J.J.: Théorème sur les déterminants, Nouv. Ann. de Math. XIII, 305. Reprinted in The collected mathematical papers of JJ Sylvester, vol. II, 28 (1854)

  35. Szegő, G.: Orthogonal Polynomials, vol. 23, 4th edn. American Mathematical Society Colloquium Publications, Providence (1975)

Download references

Acknowledgments

The authors would like to thank Prof. Erik Koelink for the helpful comment and the anonymous referee for pertinent observations and remarks. The first two authors would like to thank the Institute for Biomathematics and Biometry at the Helmholtz Zentrum München for their support during a research visit.

Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa

    A. Jooste & K. Jordaan

  2. Institute for Biomathematics and Biometry, Helmholtz Zentrum München, Neuherberg, Germany

    F. Toókos

Authors
  1. A. Jooste
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Jordaan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. Toókos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Jordaan.

Additional information

Research by K. Jordaan was partially supported by the National Research Foundation under grant number 2054423.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jooste, A., Jordaan, K. & Toókos, F. On the zeros of Meixner polynomials. Numer. Math. 124, 57–71 (2013). https://doi.org/10.1007/s00211-012-0504-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-012-0504-6

Mathematics Subject Classification (2000)

Search

Navigation