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On the Complex Zeros of Some Families of Orthogonal Polynomials

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  • Eugenia N. Petropoulou

Abstract

The complex zeros of the orthogonal Laguerre polynomials ð ¿ ð ‘› ( ð ‘Ž ) ( ð ‘¥ ) for ð ‘Ž < − ð ‘› , ultraspherical polynomials 𠑃 ð ‘› ( 𠜆 ) ( ð ‘¥ ) for 𠜆 < − ð ‘› , Jacobi polynomials 𠑃 ð ‘› ( ð ‘Ž , ð ›½ ) ( ð ‘¥ ) for ð ‘Ž < − ð ‘› , ð ›½ < − ð ‘› , ð ‘Ž + ð ›½ < − 2 ( ð ‘› + 1 ) , orthonormal Al-Salam-Carlitz II polynomials 𠑃 ð ‘› ( ð ‘Ž ) ( ð ‘¥ ; ð ‘ž ) for ð ‘Ž < 0 , 0 < ð ‘ž < 1 , and ð ‘ž -Laguerre polynomials ð ¿ ð ‘› ( ð ‘Ž ) ( ð ‘¥ ; ð ‘ž ) for ð ‘Ž < − ð ‘› , 0 < ð ‘ž < 1 are studied. Several inequalities regarding the real and imaginary properties of these zeros are given, which help locating their position. Moreover, a few limit relations regarding the asymptotic behavior of these zeros are proved. The method used is a functional analytic one. The obtained results complement and improve previously known results.

Suggested Citation

  • Eugenia N. Petropoulou, 2010. "On the Complex Zeros of Some Families of Orthogonal Polynomials," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-14, July.
    • Handle: RePEc:hin:jnlaaa:263860
    DOI: 10.1155/2010/263860
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