Monotonicity of zeros of Laguerre-Sobolev-type orthogonal polynomials
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Data
2010-08-01
Autores
Dimitrov, Dimitar Kolev [UNESP]
Marcellan, Francisco
Rafaeli, Fernando R.
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Editor
Academic Press Inc. Elsevier B.V.
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.
Descrição
Palavras-chave
Orthogonal polynomials, Laguerre polynomial, Sobolev-type orthogonal polynomials, Zeros, Monotonicity, Asymptotic
Como citar
Journal of Mathematical Analysis and Applications. San Diego: Academic Press Inc. Elsevier B.V., v. 368, n. 1, p. 80-89, 2010.