Monotonicity of zeros of Laguerre-Sobolev-type orthogonal polynomials

Dimitrov, Dimitar Kolev [UNESP]; Marcellan, Francisco; Rafaeli, Fernando R.

Monotonicity of zeros of Laguerre-Sobolev-type orthogonal polynomials

Data

2010-08-01

Autores

Dimitrov, Dimitar Kolev

Marcellan, Francisco

Rafaeli, Fernando R.

Editor

Academic Press Inc. Elsevier B.V.

Tipo

Artigo

Direito de acesso

Acesso restrito

Resumo

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 B.V. All rights reserved.