Monotonicity of zeros of Jacobi polynomials

• Autores: Dimitar K. Dimitrov, Fernando R. Rafaeli
• Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 149, Nº 1, 2007, págs. 15-29
• Idioma: inglés
• DOI: 10.1016/j.jat.2007.04.004
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• Resumen

  • Denote by xnk(a,ß), k=1,�,n, the zeros of the Jacobi polynomial . It is well known that xnk(a,ß) are increasing functions of ß and decreasing functions of a. In this paper we investigate the question of how fast the functions 1-xnk(a,ß) decrease as ß increases. We prove that the products tnk(a,ß)fn(a,ß)(1-xnk(a,ß)), where fn(a,ß)=2n2+2n(a+ß+1)+(a+1)(ß+1) are already increasing functions of ß and that, for any fixed a>-1, fn(a,ß) is the asymptotically extremal, with respect to n, function of ß that forces the products tnk(a,ß) to increase.