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, Campus Lagoas-Marcosende, 36200 Vigo, Spain;Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil;Departamento de Matemática Aplicada II, E.T.S.I. Industriales, Universidade de Vigo, Campus Lagoas-Marcosende, 36200 Vigo, Spain;Departement de Mathématique, Unité d'Analyse Mathématique et de mécanique, Université Catholique de Louvain, Bítiment Marc de Hemptinne, Chemin du Cyclotron 2, B-1348 ...

  • 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 Q"n^(^@a^,^@b^)(x), @a,@b-1, i.e. the second solution of the differential equation(1-x^2)y^''(x)+(@b-@a-(@a+@b+2)x)y^'(x)+n(n+@a+@b+1)y(x)=0,is determined for every n@?N and for all values of the parameters @a-1 and @b-1. It turns out that this number depends essentially on @a and @b as well as on the specific normalization of the function Q"n^(^@a^,^@b^)(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.