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This volume comprises the Lecture Notes of the CIMPA Summer School "Arithmetic and Geometry around Hypergeometric Functions" held at Galatasaray University, Istanbul in 2005. It contains lecture notes, a survey article, research articles, and the results of a problem session. Key topics are moduli spaces of points on P1 and Picard-Terada-Deligne-Mostow theory, moduli spaces of K3 surfaces, complex hyperbolic geometry, ball quotients, GKZ hypergeometric structures, Hilbert and Picard modular surfaces, uniformizations of complex orbifolds, algebraicity of values of Schwartz triangle functions, and Thakur's hypergeometric function. The book provides a background, gives detailed expositions and indicates new research directions. It is directed to postgraduate students and researchers.

Mathematics. --- Algebraic geometry. --- Special functions. --- Special Functions. --- Algebraic Geometry. --- Math --- Science --- Special functions --- Mathematical analysis --- Algebraic geometry --- Geometry --- Functions, special. --- Geometry, algebraic. --- Algebra --- Geometry, Algebraic --- Number theory

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Undergraduate courses in mathematics are commonly of two types. On the one hand are courses in subjects—such as linear algebra or real analysis—with which it is considered that every student of mathematics should be acquainted. On the other hand are courses given by lecturers in their own areas of specialization, which are intended to serve as a preparation for research. But after taking courses of only these two types, students might not perceive the sometimes surprising interrelationships and analogies between different branches of mathematics, and students who do not go on to become professional mathematicians might never gain a clear understanding of the nature and extent of mathematics. The two-volume Number Theory: An Introduction to Mathematics attempts to provide such an understanding of the nature and extent of mathematics. It is a modern introduction to the theory of numbers, emphasizing its connections with other branches of mathematics. Part A, which should be accessible to a first-year undergraduate, deals with elementary number theory. Part B is more advanced than the first and should give the reader some idea of the scope of mathematics today. The connecting theme is the theory of numbers. By exploring its many connections with other branches, we may obtain a broad picture. Audience This book is intended for undergraduate students in mathematics and engineering.

Number theory --- Number study --- Numbers, Theory of --- Algebra --- Number theory. --- Matrix theory. --- Functions, special. --- Number Theory. --- Linear and Multilinear Algebras, Matrix Theory. --- Special Functions. --- Special functions --- Mathematical analysis --- Algebra. --- Special functions. --- Mathematics --- Functions, Special.

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The subject of special functions is often presented as a collection of disparate results, which are rarely organised in a coherent way. This book answers the need for a different approach to the subject. The authors' main goals are to emphasise general unifying principles coherently and to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more, including chapters on discrete orthogonal polynomials and elliptic functions. The authors show how a very large part of the subject traces back to two equations - the hypergeometric equation and the confluent hypergeometric equation - and describe the various ways in which these equations are canonical and special. Providing ready access to theory and formulas, this book serves as an ideal graduate-level textbook as well as a convenient reference.

Functions, Special --- Functions, Special. --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Special functions

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This is a unique book for studying special functions through zeta-functions. Many important formulas of special functions scattered throughout the literature are located in their proper positions and readers get enlightened access to them in this book. The areas covered include: Bernoulli polynomials, the gamma function (the beta and the digamma function), the zeta-functions (the Hurwitz, the Lerch, and the Epstein zeta-function), Bessel functions, an introduction to Fourier analysis, finite Fourier series, Dirichlet L-functions, the rudiments of complex functions and summation formulas. The F

Functions, Special. --- Bernoulli polynomials. --- Bernoullian polynomials --- Bernoulli's polynomials --- Polynomials --- Sequences (Mathematics) --- Special functions --- Mathematical analysis

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This book, dedicated to Mizan Rahman, is made up of a collection of articles on various aspects of q-series and special functions. Also, it includes an article by Askey, Ismail, and Koelink on Rahman’s mathematical contributions and how they influenced the recent upsurge in the subject. Audience This book is intended for researchers and graduate students in special functions, algebraic combinatorics, quantum groups, and integrable systems.

Functions, Special. --- Special functions --- Mathematical analysis --- Functions, special. --- Mathematics. --- Integral Transforms. --- Special Functions. --- Approximations and Expansions. --- Integral Transforms, Operational Calculus. --- Transform calculus --- Integral equations --- Transformations (Mathematics) --- Math --- Science --- Special functions. --- Approximation theory. --- Integral transforms. --- Operational calculus. --- Operational calculus --- Differential equations --- Electric circuits --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems