### Abstract

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 language | English (US) |
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Pages (from-to) | 181-226 |

Number of pages | 46 |

Journal | Constructive Approximation |

Volume | 11 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 1995 |

Externally published | Yes |

### Keywords

- 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
- Mathematics(all)
- Computational Mathematics

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## Cite this

*Constructive Approximation*,

*11*(2), 181-226. https://doi.org/10.1007/BF01203415