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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
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The Table of Integrals, Series, and Products is the major reference source for integrals in the English language.It is designed for use by mathematicians, scientists, and professional engineers who need to solve complex mathematical problems.*Completely reset edition of Gradshteyn and Ryzhik reference book*New entries and sections kept in orginal numbering system with an expanded bibliography*Enlargement of material on orthogonal polynomials, theta functions, Laplace and Fourier transform pairs and much more.orthogonal polynomials, theta functions, Laplace and Fourier tr
Functional analysis --- Mathematics --- Mathématiques --- Tables. --- Tables --- 517.58 --- 517.3 --- 519.66 --- -Math --- Science --- 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. --- Integral calculus. Integration --- Mathematic tables and their compilation --- -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. --- 519.66 Mathematic tables and their compilation --- 517.3 Integral calculus. Integration --- 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. --- -519.66 Mathematic tables and their compilation --- Math --- 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 --- Logarithms. --- Logs (Logarithms) --- Algebra --- Mathematics - Tables
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The various types of special functions have become essential tools for scientists and engineers. One of the important classes of special functions is of the hypergeometric type. It includes all classical hypergeometric functions such as the well-known Gaussian hypergeometric functions, the Bessel, Macdonald, Legendre, Whittaker, Kummer, Tricomi and Wright functions, the generalized hypergeometric functions ? Fq , Meijer's G -function, Fox's H -function, etc. Application of the new special functions allows one to increase considerably the number of problems whose solutions are found in a closed
Legendre's functions. --- Spherical harmonics. --- Functions, Potential --- Potential functions --- Harmonic analysis --- Harmonic functions --- Functions, Legendre's --- Legendre's coefficients --- Legendre's equation --- Spherical harmonics --- Legendre's functions --- 517.58 --- 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. --- 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.
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The first modern treatment of orthogonal polynomials from the viewpoint of special functions is now available in paperback. Its encyclopedic coverage includes classical topics such as Jacobi, Hermite, Laguerre, Hahn, Charlier and Meixner polynomials as well as those discovered over the last 50 years, e.g. Askey-Wilson and Al-Salam-Chihara polynomial systems. Multiple orthogonal polynomials are discussed here for the first time in book form. Many modern applications of the subject are dealt with, including birth and death processes, integrable systems, combinatorics, and physical models. A chapter on open research problems and conjectures is designed to stimulate further research on the subject. Thoroughly updated and corrected since its original printing, this book continues to be valued as an authoritative reference not only by mathematicians, but also a wide range of scientists and engineers. Exercises ranging in difficulty are included to help both the graduate student and the newcomer.
Orthogonal polynomials. --- Polynômes orthogonaux --- Polynômes orthogonaux --- Orthogonal polynomials --- 517.518.8 --- 517.58 --- 517.518.8 Approximation of functions by polynomials and their generalizations --- Approximation of functions by polynomials and their generalizations --- 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. --- 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. --- Fourier analysis --- Functions, Orthogonal --- Polynomials --- 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
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Bibliothèque Houziaux
#TELE:TMIC --- 517.5 --- Functions --- Mathematics --- -510 --- Math --- Science --- Analysis (Mathematics) --- Differential equations --- Mathematical analysis --- Numbers, Complex --- Set theory --- Calculus --- Theory of functions --- Tables --- 517.5 Theory of functions --- 519.66 --- #TELE:d.d. Prof. A. J. J. Oosterlinck --- 519.66 Mathematic tables and their compilation --- Mathematic tables and their compilation --- 681.3 *G10 --- #TWER:BOEK --- 51 --- functies --- tabellenboeken --- Computerwetenschap--?*G10 --- Wiskunde --- 517.58 --- #ABIB:astp --- 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. --- 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. --- #TELE:MI2 --- functies (wiskunde) --- integralen --- Functions. --- Tables. --- Fonctions (Mathématiques) --- Mathématiques --- Fonctions (mathématiques). --- Analyse numérique. --- Numerical analysis --- Analyse mathématique --- Analyse combinatoire --- Systèmes d'aide-mémoire --- Mathematics - Tables --- Fonctions speciales --- Formulaire de mathematiques --- Algebras --- Constants --- Probability theory --- -Tables --- Fonction mathematique
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In 1922, Harald Bohr and Johannes Mollerup established a remarkable characterization of the Euler gamma function using its log-convexity property. A decade later, Emil Artin investigated this result and used it to derive the basic properties of the gamma function using elementary methods of the calculus. Bohr-Mollerup's theorem was then adopted by Nicolas Bourbaki as the starting point for his exposition of the gamma function. This open access book develops a far-reaching generalization of Bohr-Mollerup's theorem to higher order convex functions, along lines initiated by Wolfgang Krull, Roger Webster, and some others but going considerably further than past work. In particular, this generalization shows using elementary techniques that a very rich spectrum of functions satisfy analogues of several classical properties of the gamma function, including Bohr-Mollerup's theorem itself, Euler's reflection formula, Gauss' multiplication theorem, Stirling's formula, and Weierstrass' canonical factorization. The scope of the theory developed in this work is illustrated through various examples, ranging from the gamma function itself and its variants and generalizations (q-gamma, polygamma, multiple gamma functions) to important special functions such as the Hurwitz zeta function and the generalized Stieltjes constants. This volume is also an opportunity to honor the 100th anniversary of Bohr-Mollerup's theorem and to spark the interest of a large number of researchers in this beautiful theory.
Convex functions. --- Gamma functions. --- Functions, Convex --- Functions of real variables --- Functions, Gamma --- Transcendental functions --- Difference Equation --- Higher Order Convexity --- Bohr-Mollerup's Theorem --- Principal Indefinite Sums --- Gauss' Limit --- Euler Product Form --- Raabe's Formula --- Binet's Function --- Stirling's Formula --- Euler's Infinite Product --- Euler's Reflection Formula --- Weierstrass' Infinite Product --- Gauss Multiplication Formula --- Euler's Constant --- Gamma Function --- Polygamma Functions --- Hurwitz Zeta Function --- Generalized Stieltjes Constants
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Special functions and orthogonal polynomials in particular have been around for centuries. Can you imagine mathematics without trigonometric functions, the exponential function or polynomials? In the twentieth century the emphasis was on special functions satisfying linear differential equations, but this has now been extended to difference equations, partial differential equations and non-linear differential equations. The present set of lecture notes containes seven chapters about the current state of orthogonal polynomials and special functions and gives a view on open problems and future directions. The topics are: computational methods and software for quadrature and approximation, equilibrium problems in logarithmic potential theory, discrete orthogonal polynomials and convergence of Krylov subspace methods in numerical linear algebra, orthogonal rational functions and matrix orthogonal rational functions, orthogonal polynomials in several variables (Jack polynomials) and separation of variables, a classification of finite families of orthogonal polynomials in Askey’s scheme using Leonard pairs, and non-linear special functions associated with the Painlevé equations.
Orthogonal polynomials. --- Functions, Special. --- Polynômes orthogonaux --- Fonctions spéciales --- Orthogonal polynomials --- Functions, Special --- Operations Research --- Mathematical Theory --- Civil & Environmental Engineering --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- 517.518.8 --- 517.58 --- Approximation of functions by polynomials and their generalizations --- 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. --- 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. --- 517.518.8 Approximation of functions by polynomials and their generalizations --- Special functions --- Mathematics. --- Approximation theory. --- Fourier analysis. --- Special functions. --- Numerical analysis. --- Approximations and Expansions. --- Special Functions. --- Numerical Analysis. --- Fourier Analysis. --- Fourier analysis --- Functions, Orthogonal --- Polynomials --- Mathematical analysis --- 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 --- Analysis, Fourier --- Theory of approximation --- Functional analysis --- Functions --- Chebyshev systems --- Math --- Science --- Functions, special.
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This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention.
bessel function --- harmonically convex function --- non-singular function involving kernel fractional operator --- Hadamard inequality --- Fejér–Hadamard inequality --- Elzaki transform --- Caputo fractional derivative --- AB-fractional operator --- new iterative transform method --- Fisher’s equation --- Hukuhara difference --- Atangana–Baleanu fractional derivative operator --- Mittag–Leffler kernel --- Fornberg–Whitham equation --- fractional div-curl systems --- Helmholtz decomposition theorem --- Riemann–Liouville derivative --- Caputo derivative --- fractional vector operators --- weighted (k,s) fractional integral operator --- weighted (k,s) fractional derivative --- weighted generalized Laplace transform --- fractional kinetic equation --- typhoid fever disease --- vaccination --- model calibration --- asymptotic stability --- fixed point theory --- nonlinear models --- efficiency index --- computational cost --- Halley’s method --- basin of attraction --- computational order of convergence --- Caputo–Hadamard fractional derivative --- thermostat modeling --- Caputo–Hadamard fractional integral --- hybrid Caputo–Hadamard fractional differential equation and inclusion --- prey-predator model --- boundedness --- period-doubling bifurcation --- Neimark-Sacker bifurcation --- hybrid control --- fractal dimensions --- cubic B-splines --- trigonometric cubic B-splines --- extended cubic B-splines --- Caputo–Fabrizio derivative --- Cattaneo equation --- Hermite-Hadamard-type inequalities --- Hilfer fractional derivative --- Hölder’s inequality --- fractional-order differential equations --- operational matrices --- shifted Vieta–Lucas polynomials --- Adomian decomposition method --- system of Whitham-Broer-Kaup equations --- Caputo-Fabrizio derivative --- Yang transform --- ϑ-Caputo derivative --- extremal solutions --- monotone iterative method --- sequences --- convex --- exponential convex --- fractional --- quantum --- inequalities --- Gould-Hopper-Laguerre-Sheffer matrix polynomials --- quasi-monomiality --- umbral calculus --- fractional calculus --- Euler’s integral of gamma functions --- beta function --- generalized hypergeometric series --- operational methods --- delta function --- Riemann zeta-function --- fractional transforms --- Fox–Wright-function --- generalized fractional kinetic equation --- n/a --- Fejér-Hadamard inequality --- Fisher's equation --- Atangana-Baleanu fractional derivative operator --- Mittag-Leffler kernel --- Fornberg-Whitham equation --- Riemann-Liouville derivative --- Halley's method --- Caputo-Hadamard fractional derivative --- Caputo-Hadamard fractional integral --- hybrid Caputo-Hadamard fractional differential equation and inclusion --- Hölder's inequality --- shifted Vieta-Lucas polynomials --- Euler's integral of gamma functions --- Fox-Wright-function
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