## The Special Functions and Their ApproximationsA detailed and self-contained and unified treatment of many mathematical functions which arise in applied problems, as well as the attendant mathematical theory for their approximations. many common features of the Bessel functions, Legendre functions, incomplete gamma functions, confluent hypergeometric functions, as well as of otherw, can be derived. Hitherto, many of the material upon which the volumes are based has been available only in papers scattered throughout the literature. |

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### Contents

1 | |

8 | |

Chapter III Hypergeometric Functions | 38 |

Chapter IV Confluent Hypergeometric Functions | 115 |

Chapter V The Generalized Hypergeometric Function and the GFunction | 136 |

Chapter VI Identification of the pFq and GFunctions with the Special Functions of Mathematical Physics | 209 |

Chapter VII Asymptotic Expansions of pFq for Large Parameters | 235 |

Chapter VIII Orthogonal Polynomials | 267 |

Bibliography | 330 |

339 | |

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### Common terms and phrases

Abramowitz and Stegun analytic continuation arbitrary integer arg z asymptotic expansion Bernoulli polynomials Bessel functions change of notation Chapter Chebyshev polynomials coefficients confluent hypergeometric functions converges Cºp deduced defined denominator parameter derived differential equation Erdélyi evaluation expansions in series exponential expressed expſin fundamental solutions G-function gamma functions given hypergeometric series incomplete gamma functions infinity integer or zero integrand Jacobi polynomial latter Lemma logarithmic solutions Math negative integer Nörlund Note numbers orthogonal path of integration poles proof proved rational approximations recursion formula relations replaced representation respectively restriction right-hand side s a positive integer s m s q satisfy series of Chebyshev singularities ſº special functions sufficiently large suppose Theorem Ti(x tºº valid values