## Special Functions |

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### Contents

INFINITE PRODUCTS Pag 1 Introduction Definition of an infinite product | 1 |

A necessary condition for convergence The associated series of logarithms | 2 |

Absolute convergence | 3 |

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### Common terms and phrases

absolutely convergent an(x asymptotic expansion basic table Bessel functions Bessel polynomials Brafman Chapter coefficients conclude Consider constant contiguous function relations defined denominator parameter derive differential equation differential recurrence relation elementary elliptic function en(x evaluate exists exp(2xt finite follows function f(z Gamma function Gegenbauer polynomials Hence Hermite polynomials hypergeometric function hypergeometric series infinite product integral Jacobi polynomials Laguerre polynomials Laplace transform left member Legendre polynomials log T(z logarithmic Math negative integer nomials non-negative integer notation obtain on(x orthogonal polynomials pn(x poles poly polynomial of degree polynomial sets positive integer power series preceding section Proof properties pure recurrence relation Rainville Re(a Re(b Re(c Re(z region replace result in Ex right member satisfied set of polynomials Sheffer A-type zero simple set singular points sn(u sn(x solution summation term Theorem 20 theta functions variable Watson Watson's lemma Weierstrass write yields