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Algebraic geometry --- Number theory --- 51 <082.1> --- Mathematics--Series --- Hypergeometric series. --- Functions, Zeta. --- Transcendental numbers. --- Séries hypergéométriques. --- Fonctions zêta. --- Nombres transcendants. --- Functions, Zeta --- Hypergeometric series --- Transcendental numbers --- Numbers, Transcendental --- Irrational numbers --- Gaussian hypergeometric series --- Gaussian series --- Gauss's series --- Series --- Hypergeometric functions --- Zeta functions

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Hypergeometric series --- Cohen-Macaulay modules --- Modules (Algebra) --- Singularities (Mathematics) --- Séries hypergéométriques --- Modules de Cohen-Macaulay --- Modules (Algèbre) --- Singularités (Mathématiques) --- Hypergeometric series. --- Cohen-Macaulay modules. --- Geometry, Algebraic --- Finite number systems --- Modular systems (Algebra) --- Algebra --- Finite groups --- Rings (Algebra) --- Macaulay modules, Cohen --- -Modules (Algebra) --- Gaussian hypergeometric series --- Gaussian series --- Gauss's series --- Series --- Hypergeometric functions --- Séries hypergéométriques. --- -Gaussian hypergeometric series --- Séries hypergéométriques --- Modules (Algèbre) --- Singularités (Mathématiques)

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This revised and expanded new edition will continue to meet the needs for an authoritative, up-to-date, self contained, and comprehensive account of the rapidly growing field of basic hypergeometric series, or q-series. Simplicity, clarity, deductive proofs, thoughtfully designed exercises, and useful appendices are among its strengths. The first five chapters cover basic hypergeometric series and integrals, whilst the next five are devoted to applications in various areas including Askey-Wilson integrals and orthogonal polynomials, partitions in number theory, multiple series, orthogonal polynomials in several variables, and generating functions. Chapters 9-11 are new for the second edition, the final chapter containing a simplified version of the main elements of the theta and elliptic hypergeometric series as a natural extension of the single-base q-series. Some sections and exercises have been added to reflect recent developments, and the Bibliography has been revised to maintain its comprehensiveness.

517.58 --- Special functions. Hyperbolic functions. Euler integrals. Gamma functions. Elliptic functions and integrals. Bessel functions. Other cylindrical functions. Spherical functions. Legendre polynomials. Orthogonal polynomials. Chebyshev polynomials. --- Hypergeometric series. --- 517.58 Special functions. Hyperbolic functions. Euler integrals. Gamma functions. Elliptic functions and integrals. Bessel functions. Other cylindrical functions. Spherical functions. Legendre polynomials. Orthogonal polynomials. Chebyshev polynomials. --- Hypergeometric series --- Gaussian hypergeometric series --- Gaussian series --- Gauss's series --- Series --- Hypergeometric functions --- Special functions. Hyperbolic functions. Euler integrals. Gamma functions. Elliptic functions and integrals. Bessel functions. Other cylindrical functions. Spherical functions. Legendre polynomials. Orthogonal polynomials. Chebyshev polynomials --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis