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Classical orthogonal polynomials as moments

  Published:1997-06-01
 Printed: Jun 1997
  • Mourad E. H. Ismail
  • Dennis Stanton
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Abstract

We show that the Meixner, Pollaczek, Meixner-Pollaczek, the continuous $q$-ultraspherical polynomials and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures. We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials.
Keywords: Classical orthogonal polynomials, \ACP, continuous, $q$-ultraspherical polynomials, generating functions, multilinear, generating functions, transformation formulas, umbral calculus Classical orthogonal polynomials, \ACP, continuous, $q$-ultraspherical polynomials, generating functions, multilinear, generating functions, transformation formulas, umbral calculus
MSC Classifications: 33D45, 33D20, 33C45, 30E05 show english descriptions Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
unknown classification 33D20
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
Moment problems, interpolation problems
33D45 - Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33D20 - unknown classification 33D20
33C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
30E05 - Moment problems, interpolation problems
 

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