Monotonicity of Zeros of Laguerre-Sobolev Type Orthogonal Polynomials Articles uri icon

publication date

  • August 2010

start page

  • 80

end page

  • 89

issue

  • 1

volume

  • 368

international standard serial number (ISSN)

  • 0022-247X

electronic international standard serial number (EISSN)

  • 1096-0813

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).