Monotonicity of zeros of Laguerre-Sobolev-type orthogonal polynomials

Author(s)

Dimitrov, Dimitar K. ^[1] ; Marcellan, Francisco ^[2] ; Rafaeli, Fernando R. ^[3]

Affiliation

^[1] Univ Estadual Paulista, IBILCE, Dept Ciencias Comp & Estatist, Sao Paulo - Brazil

^[2] Univ Carlos III, Escuela Politecn Super, Dept Matemat, Leganes - Spain

^[3] Univ Estadual Campinas, Inst Matemat Estatist & Comp, BR-13081970 Campinas, SP - Brazil

• Document type: Journal article
• Source: Journal of Mathematical Analysis and Applications; v. 368, n. 1, p. 80-89, AUG 1 2010.
• Web of Science Citations: 13

Abstract

Denote by x(n,k)(M,N)(alpha), k = 1, ..., n, the zeros of the Laguerre-Sobolev-type polynomials L(n)((alpha, M, N))(x) orthogonal with respect to the inner product < p, q > = 1/Gamma(alpha + 1) integral(infinity)(0)p(x)q(x)x(alpha)e(-x) dx + Mp(0)q(0) + Np'(0)q'(0), where alpha > -1, M >= 0 and N >= 0. We prove that x(n,k)(M,N)(alpha) interlace with the zeros of Laguerre orthogonal polynomials L(n)((alpha))(x) and establish monotonicity with respect to the parameters M and N of x(n,k)(M,0)(alpha) and x(n,k)(0,N)(alpha). Moreover, we find N(0) such that x(n,n)(M,N)(alpha) < 0 for all N > N(0), where x(n,n)(M,N)(alpha) is the smallest zero of L(n)((alpha, M, N))(x). Further, we present monotonicity and asymptotic relations of certain functions involving x(n,k)(M,0)(alpha) and x(n,k)(0,N)(alpha). (C) 2010 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 03/01874-2 - Orthogonal and similar polynomials: properties and applications
• Grantee: Alagacone Sri Ranga
• Support Opportunities: Research Projects - Thematic Grants