Orthogonal Polynomials with Ultra-Exponential Weight Functions: An Explicit Solution to the Ditkin–Prudnikov Problem

Abstract

New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, we give an explicit solution to the Ditkin–Prudnikov problem (1966). The 3-term recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are derived. The method is based on differential properties of the involved special functions and their representations in terms of the Mellin–Barnes and Laplace integrals. A notion of the composition polynomial orthogonality is introduced. The corresponding advantages of this orthogonality to discover new sequences of polynomials and their relations to the corresponding multiple orthogonal polynomial ensembles are shown.

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Correspondence to S. Yakubovich.

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The work was partially supported by CMUP, which is financed by national funds through FCT (Portugal) under the project with reference UIDB/00144/2020. The author thanks Marco Martins Afonso for necessary numerical calculations and verifications of some formulas. Finally, the author is sincerely indebted to referees for useful comments and suggestions which rather improved the presentation of the paper.

Communicated by Erik Koelink.

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Yakubovich, S. Orthogonal Polynomials with Ultra-Exponential Weight Functions: An Explicit Solution to the Ditkin–Prudnikov Problem. Constr Approx 53, 1–38 (2021). https://doi.org/10.1007/s00365-020-09523-0

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Keywords

  • Orthogonal polynomials
  • Modified Bessel functions
  • Meijer G-function
  • Mellin transform
  • Associated Laguerre polynomials
  • Multiple orthogonal polynomials

Mathematics Subject Classification

  • 33C47
  • 33C45
  • 33C10
  • 44A15
  • 42C05