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A conjecture on the zeros of ultraspherical polynomials
Journal of Computational and Applied Mathematics 133 (2001) 684 www.elsevier.com/locate/cam
A conjecture on the zeros of ultraspherical polynomials a
% ad Elberta; X , Andrea Laforgiab , Panayiotis Siafarikasc Arp% Mathematical Institute, Hungarian Academy of Sciences, P.O Box 127, H-1364 Budapest, Hungary b Dipartimento di Matematica, Universit(a di Roma Tre, Largo S. Murialdo 1, I-00146 Roma, Italy c Department of Mathematics, University of Patras, 26500 Patras, Greece
For k = 1; 2; : : : ; [n=2] and ¿ − 12 , let x() nk be the kth positive zero, in decreasing order, of the () ultraspherical polynomial Pn (x), of degree n; n = 1; 2; : : : . We formulate the following conjecture: x() nk is a convex function of : Remark. Elbert and Laforgia [1] proved that @2 () 3 x = hn; k ; →∞ @2 nk 4 where hn; k is the kth zero of the Hermite polynomial Hn (x), of degree n. √ Kokologiannaki and Siafarikas [2] proved the conjecture under the restriction ¿ n= 3 + only for k = 1. lim 5=2
1 2
and
References % Elbert, A. Laforgia, Asymptotic formulas for ultraspherical polynomials Pn() (x) and their zeros, for large values [1] A. of , Proc. Amer. Math. Soc. 114 (1992) 371–377. [2] C.G. Kokologiannaki, P. Siafarikas, Convexity of the largest zero of the ultraspherical polynomials, Integral Transforms Special Funct. 4 (1996) 1–6.
X Deceased on 25 April 2001.
E-mail address: [email protected] (A. Laforgia), [email protected] (P. Siafarikas). c 2001 Elsevier Science B.V. All rights reserved. 0377-0427/01/$ - see front matter PII: S 0 3 7 7 - 0 4 2 7 ( 0 0 ) 0 0 7 1 8 - 4
A conjecture on the zeros of ultraspherical polynomials a
% ad Elberta; X , Andrea Laforgiab , Panayiotis Siafarikasc Arp% Mathematical Institute, Hungarian Academy of Sciences, P.O Box 127, H-1364 Budapest, Hungary b Dipartimento di Matematica, Universit(a di Roma Tre, Largo S. Murialdo 1, I-00146 Roma, Italy c Department of Mathematics, University of Patras, 26500 Patras, Greece
For k = 1; 2; : : : ; [n=2] and ¿ − 12 , let x() nk be the kth positive zero, in decreasing order, of the () ultraspherical polynomial Pn (x), of degree n; n = 1; 2; : : : . We formulate the following conjecture: x() nk is a convex function of : Remark. Elbert and Laforgia [1] proved that @2 () 3 x = hn; k ; →∞ @2 nk 4 where hn; k is the kth zero of the Hermite polynomial Hn (x), of degree n. √ Kokologiannaki and Siafarikas [2] proved the conjecture under the restriction ¿ n= 3 + only for k = 1. lim 5=2
1 2
and
References % Elbert, A. Laforgia, Asymptotic formulas for ultraspherical polynomials Pn() (x) and their zeros, for large values [1] A. of , Proc. Amer. Math. Soc. 114 (1992) 371–377. [2] C.G. Kokologiannaki, P. Siafarikas, Convexity of the largest zero of the ultraspherical polynomials, Integral Transforms Special Funct. 4 (1996) 1–6.
X Deceased on 25 April 2001.
E-mail address: [email protected] (A. Laforgia), [email protected] (P. Siafarikas). c 2001 Elsevier Science B.V. All rights reserved. 0377-0427/01/$ - see front matter PII: S 0 3 7 7 - 0 4 2 7 ( 0 0 ) 0 0 7 1 8 - 4