Geodesic
On the discrete extension of Markov's theorem on monotonicity of zeros
Electronic transactions on numerical analysis, Tome 44 (2015), pp. 271-280
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 $1/c$.
Classification : 33C45, 30C15
Keywords: orthogonal polynomials on the real line, Uvarov's transformation, Markov's theorem, monotonicity of zeros, asymptotic behavior, speed of convergence 
@article{ETNA_2015__44__a17,
     author = {Castillo,  Kenier and Rafaeli,  Fernando R.},
     title = {On the discrete extension of {Markov's} theorem on monotonicity of zeros},
     journal = {Electronic transactions on numerical analysis},
     pages = {271--280},
     year = {2015},
     volume = {44},
     zbl = {1333.33013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2015__44__a17/}
}
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Castillo,  Kenier; Rafaeli,  Fernando R. On the discrete extension of Markov's theorem on monotonicity of zeros. Electronic transactions on numerical analysis, Tome 44 (2015), pp. 271-280. http://geodesic.mathdoc.fr/item/ETNA_2015__44__a17/