Zeros of Jacobi functions of second kind

  • Authors:

  • Iván Area;Dimitar K. Dimitrov;Eduardo Godoy;André Ronveaux

  • Affiliations:

  • Departamento de Matemática Aplicada II, E.T.S.E. Telecomunicación, Universidade de Vigo, Vigo, Spain;Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, São José do Rio Preto, SP, Brazil;Departamento de Matemática Aplicada II, E.T.S.I Industriales, Universidade de Vigo, Vigo, Spain;Departement de Mathématique, Unité d'Analyse Mathématique et de mécanique, Université Catholique de Louvain, Louvain-la-Neuve, Belgium

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2006

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Abstract

The number of zeros in (-1,1) of the Jacobi function of second kind Qn(α, β) (x), α, β -1, i.e. the second solution of the differential equation (1 - x2)y"(x) + (β - α - (α + β + 2)x)y'(x) + n(n + α + β + 1)y(x) = 0, is determined for every n ∈ N and for all values of the parameters α -1 and β -1. It turns out that this number depends essentially on α and β as well as on the specific normalization of the function Qn(α, β) (x). Interlacing properties of the zeros are also obtained. As a consequence of the main result, we determine the number of zeros of Laguerre's and Hermite's functions of second kind.