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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | Print version: Doman, Brian George Spencer. Classical Orthogonal Polynomials. Singapore : World Scientific Publishing Company, ©2015 |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
Brian George Spencer Doman |

ISBN: | 9789814704045 9814704040 9789814704038 9814704032 |

OCLC Number: | 944466902 |

Description: | 1 online resource (xii, 164 pages) |

Contents: | Preface; 1. Definitions and General Properties; 1.1 Introduction; 1.2 Definition of Orthogonality; 1.3 Gram-Schmidt Orthogonalisation Procedure; 1.4 The Nth Order Polynomial RN(x) has N Distinct Real Zeros in the Interval (a, b); 1.5 Gauss Quadrature Formula; 1.6 Recurrence Relation; 1.7 Favards Theorem; 1.8 The Christoffel-Darboux Formula; 1.9 Interlacing of Zeros; 1.10 Minimum Property; 1.11 Approximation of Functions; 1.12 Definitions of Some Parameters; 1.13 Fundamental Intervals and Weight Functions of the Classical Orthogonal Polynomials; 1.14 Recurrence Relations. 1.15 Differential Relation1.16 Step Up and Step Down Operators; 1.17 Interlacing of Zeros; 1.18 Differential Equation; References; 2. Hermite Polynomials; 2.1 Introduction; 2.2 Differential Equation; 2.3 Orthogonality; 2.4 Derivative Property; 2.5 Rodrigues Formula; 2.6 Explicit Expression; 2.7 Generating Function; 2.8 Recurrence Relations; 2.9 Addition Formulae; 2.10 Step Up and Step Down Operators; 2.11 Parabolic Cylinder Functions; References; 3. Associated Laguerre Polynomials; 3.1 Introduction; 3.2 Differential Equation; 3.3 Orthogonality; 3.4 Derivative Property; 3.5 Rodrigues Formula. 3.6 Explicit Expression3.7 Generating Function; 3.8 Recurrence Relations; 3.9 Addition Formulae; 3.10 Differential Relations; 3.11 Step Up and Step Down Operators; References; 4. Legendre Polynomials; 4.1 Introduction; 4.2 Differential Equation; 4.3 Orthogonality; 4.4 Rodrigues Formula; 4.5 Explicit Expression; 4.6 Generating Function; 4.7 Recurrence Relations; 4.8 Differential Relation; 4.9 Step Up and Step Down Operators; 4.10 Appendix; References; 5. Chebyshev Polynomials of the First Kind; 5.1 Introduction; 5.2 Differential Equation; 5.3 Orthogonality; 5.4 Trigonometric Representation. 5.5 Explicit Expression5.6 Rodrigues Formula; 5.7 Generating Functions; 5.8 Recurrence Relations; 5.9 Addition Formulae; 5.10 Differential Relations; 5.11 Relations with Other Chebyshev Polynomials; 5.12 Step Up and Step Down Operators; References; 6. Chebyshev Polynomials of the Second Kind; 6.1 Introduction; 6.2 Differential Equation; 6.3 Orthogonality; 6.4 Trigonometric Representation; 6.5 Explicit Expression; 6.6 Rodrigues Formula; 6.7 Generating Functions; 6.8 Recurrence Relations; 6.9 Addition Formula; 6.10 Differential Relations; 6.11 Step Up and Step Down Operators; References. 7. Chebyshev Polynomials of the Third Kind7.1 Introduction; 7.2 Differential Equation; 7.3 Orthogonality; 7.4 Trigonometric Representaion; 7.5 Rodrigues Formula; 7.6 Explicit Expression; 7.7 Generating Functions; 7.8 Recurrence Relations; 7.9 Differential Relation; 7.10 Step Up and Step Down Operators; References; 8. Chebyshev Polynomials of the Fourth Kind; 8.1 Introduction; 8.2 Differential Equation; 8.3 Orthogonality; 8.4 Trigonometric Representation; 8.5 Rodrigues Formula; 8.6 Explicit Expression; 8.7 Generating Functions; 8.8 Recurrence Relations; 8.9 Differential Relation. |

Responsibility: | Brian George Spencer Doman (University of Liverpool, UK). |

### Abstract:

"This book defines sets of orthogonal polynomials and derives a number of properties satisfied by any such set. It continues by describing the classical orthogonal polynomials and the additional properties they have. The first chapter defines the orthogonality condition for two functions. It then gives an iterative process to produce a set of polynomials which are orthogonal to one another and then describes a number of properties satisfied by any set of orthogonal polynomials. The classical orthogonal polynomials arise when the weight function in the orthogonality condition has a particular form. These polynomials have a further set of properties and in particular satisfy a second order differential equation. Each subsequent chapter investigates the properties of a particular polynomial set starting from its differential equation."--

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