### 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) |
---|---|

Pages (from-to) | 181-226 |

Number of pages | 46 |

Journal | Constructive Approximation |

Volume | 11 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1995 |

Externally published | Yes |

### Fingerprint

### 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
- q-linear
- q-quadratic lattices
- quadratic
- Rodrigues formula

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)

### Cite this

*Constructive Approximation*,

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

**On classical orthogonal polynomials.** / Atakishiyev, N. M.; Rahman, M.; Suslov, Sergei.

Research output: Contribution to journal › Article

*Constructive Approximation*, vol. 11, no. 2, pp. 181-226. https://doi.org/10.1007/BF01203415

}

TY - JOUR

T1 - On classical orthogonal polynomials

AU - Atakishiyev, N. M.

AU - Rahman, M.

AU - Suslov, Sergei

PY - 1995/6

Y1 - 1995/6

N2 - 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.

AB - 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.

KW - AMS classification: Primary 33D20, 33D45, Secondary 33C45

KW - Andrews-Askey definition

KW - Classical orthogonal polynomials

KW - Hahn's characterization

KW - Hypergeometric-type difference equation

KW - Integral and series representations of discrete and continuous classical orthogonal polynomials

KW - Linear

KW - Moments

KW - q-linear

KW - q-quadratic lattices

KW - quadratic

KW - Rodrigues formula

UR - http://www.scopus.com/inward/record.url?scp=21844496471&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21844496471&partnerID=8YFLogxK

U2 - 10.1007/BF01203415

DO - 10.1007/BF01203415

M3 - Article

AN - SCOPUS:21844496471

VL - 11

SP - 181

EP - 226

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

IS - 2

ER -