Volume 44, pp. 271-280, 2015.

On the discrete extension of Markov's theorem on monotonicity of zeros

Kenier Castillo and Fernando R. Rafaeli


Motivated by an open problem proposed by M. E. H. Ismail in his monograph “Classical and quantum orthogonal polynomials in one variable” (Cambridge University Press, 2005), we study the behavior of zeros of orthogonal polynomials associated with a positive measure on $[a,b] \subseteq \mathbb{R}$ which is modified by adding a mass at $c\in \mathbb{R} \setminus (a,b)$. We prove that the zeros of the corresponding polynomials are strictly increasing functions of $c$. Moreover, we establish their asymptotics when $c$ tends to infinity or minus infinity, and it is shown that the rate of convergence is of order $1/c$.

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Key words

orthogonal polynomials on the real line, Uvarov's transformation, Markov's theorem, monotonicity of zeros, asymptotic behavior, speed of convergence

AMS subject classifications

33C45, 30C15

Links to the cited ETNA articles

[6]Vol. 38 (2011), pp. 317-326 K. Driver, A. Jooste, and K. Jordaan: Stieltjes interlacing of zeros of Jacobi polynomials from different sequences
[7]Vol. 19 (2005), pp. 37-47 A. Garrido, J. Arvesú, and F. Marcellán: An electrostatic interpretation of the zeros of the Freud-type orthogonal polynomials

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