New bounds for the extreme zeros of Jacobi polynomials

Nikolov, Geno

doi:10.1090/proc/14370

New bounds for the extreme zeros of Jacobi polynomials
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by Geno Nikolov

Proc. Amer. Math. Soc. 147 (2019), 1541-1550

DOI: https://doi.org/10.1090/proc/14370

Published electronically: January 8, 2019

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Abstract:

We apply the Euler–Rayleigh method to the Jacobi and, in particular, the Gegenbauer polynomials, represented as hypergeometric functions, to prove new bounds for the extreme zeroes of these polynomials. Our bounds are shown to either reproduce or improve some of the recent results obtained by other authors.

References

Dragomir Aleksov and Geno Nikolov, Markov $L_2$ inequality with the Gegenbauer weight, J. Approx. Theory 225 (2018), 224–241. MR 3733258, DOI 10.1016/j.jat.2017.10.008
Iván Area, Dimitar K. Dimitrov, Eduardo Godoy, and André Ronveaux, Zeros of Gegenbauer and Hermite polynomials and connection coefficients, Math. Comp. 73 (2004), no. 248, 1937–1951. MR 2059744, DOI 10.1090/S0025-5718-04-01642-4
Iván Area, Dimitar K. Dimitrov, Eduardo Godoy, and Fernando R. Rafaeli, Inequalities for zeros of Jacobi polynomials via Obrechkoff’s theorem, Math. Comp. 81 (2012), no. 278, 991–1004. MR 2869046, DOI 10.1090/S0025-5718-2011-02553-6
Dimitar K. Dimitrov and Geno P. Nikolov, Sharp bounds for the extreme zeros of classical orthogonal polynomials, J. Approx. Theory 162 (2010), no. 10, 1793–1804. MR 2728047, DOI 10.1016/j.jat.2009.11.006
P. Dörfler, Asymptotics of the best constant in a certain Markov-type inequality, J. Approx. Theory 114 (2002), no. 1, 84–97. MR 1880296, DOI 10.1006/jath.2001.3638
K. Driver and K. Jordaan, Bounds for extreme zeros of some classical orthogonal polynomials, J. Approx. Theory 164 (2012), no. 9, 1200–1204. MR 2948561, DOI 10.1016/j.jat.2012.05.014
K. Driver and K. Jordaan, Inequalities for extreme zeros of some classical orthogonal and $q$-orthogonal polynomials, Math. Model. Nat. Phenom. 8 (2013), no. 1, 48–59. MR 3022978, DOI 10.1051/mmnp/20138103
Klaus-Jürgen Förster and Knut Petras, On estimates for the weights in Gaussian quadrature in the ultraspherical case, Math. Comp. 55 (1990), no. 191, 243–264. MR 1023758, DOI 10.1090/S0025-5718-1990-1023758-1
Dharma P. Gupta and Martin E. Muldoon, Inequalities for the smallest zeros of Laguerre polynomials and their $q$-analogues, JIPAM. J. Inequal. Pure Appl. Math. 8 (2007), no. 1, Article 24, 7. MR 2295718
Mourand E. H. Ismail and Xin Li, Bound on the extreme zeros of orthogonal polynomials, Proc. Amer. Math. Soc. 115 (1992), no. 1, 131–140. MR 1079891, DOI 10.1090/S0002-9939-1992-1079891-5
Mourad E. H. Ismail and Martin E. Muldoon, Bounds for the small real and purely imaginary zeros of Bessel and related functions, Methods Appl. Anal. 2 (1995), no. 1, 1–21. MR 1337450, DOI 10.4310/MAA.1995.v2.n1.a1
Ilia Krasikov, Bounds for zeros of the Laguerre polynomials, J. Approx. Theory 121 (2003), no. 2, 287–291. MR 1971774, DOI 10.1016/S0021-9045(03)00029-7
Geno Nikolov and Alexei Shadrin, On the $L_2$ Markov inequality with Laguerre weight, Progress in approximation theory and applicable complex analysis, Springer Optim. Appl., vol. 117, Springer, Cham, 2017, pp. 1–17. MR 3644736
Geno Nikolov and Rumen Uluchev, Estimates for the best constant in a Markov $L_2$–inequality with the assistance of computer algebra, Ann. Univ. Sofia, Ser. Math. Inf. 104 (2017), 55–75.
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, RI, 1975. MR 372517
Erik A. van Doorn, Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices, J. Approx. Theory 51 (1987), no. 3, 254–266. MR 913621, DOI 10.1016/0021-9045(87)90038-4
B. L. van der Waerden, Modern Algebra. Vol. I, Frederick Ungar Publishing Co., New York, 1949. Translated from the second revised German edition by Fred Blum; With revisions and additions by the author. MR 29363