On classical orthogonal polynomials

N. M. Atakishiyev, M. Rahman, S. K. Suslov

Research output: Contribution to journalArticlepeer-review

69 Scopus citations


Following the works of Nikiforov and Uvarov a review of the hypergeometric-type difference equation for a function y(x(s)) on a nonuniform lattice x(s) is given. It is shown that the difference-derivatives of y(x(s)) also satisfy similar equations, if and only if x(s) is a linear, q-linear, quadratic, or a q-quadratic lattice. This characterization is then used to give a definition of classical orthogonal polynomials, in the broad sense of Hahn, and consistent with the latest definition proposed by Andrews and Askey. The rest of the paper is concerned with the details of the solutions: orthogonality, boundary conditions, moments, integral representations, etc. A classification of classical orthogonal polynomials, discrete as well as continuous, on the basis of lattice type, is also presented.

Original languageEnglish (US)
Pages (from-to)181-226
Number of pages46
JournalConstructive Approximation
Issue number2
StatePublished - Jun 1 1995
Externally publishedYes


  • AMS classification: Primary 33D20, 33D45, Secondary 33C45
  • Andrews-Askey definition
  • Classical orthogonal polynomials
  • Hahn's characterization
  • Hypergeometric-type difference equation
  • Integral and series representations of discrete and continuous classical orthogonal polynomials
  • Linear
  • Moments
  • Rodrigues formula
  • q-linear
  • q-quadratic lattices
  • quadratic

ASJC Scopus subject areas

  • Analysis
  • General Mathematics
  • Computational Mathematics


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