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The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between integer-order operators. This field includes classical fractional operators such as Riemann–Liouville, Weyl, Caputo, and Grunwald–Letnikov; nevertheless, especially in the last two decades, many new operators have also appeared that often define using integrals with special functions in the kernel, such as Atangana–Baleanu, Prabhakar, Marichev–Saigo–Maeda, and the tempered fractional equation, as well as their extended or multivariable forms. These have been intensively studied because they can also be useful in modelling and analysing real-world processes, due to their different properties and behaviours from those of the classical cases.Special functions, such as Mittag–Leffler functions, hypergeometric functions, Fox's H-functions, Wright functions, and Bessel and hyper-Bessel functions, also have important connections with fractional calculus. Some of them, such as the Mittag–Leffler function and its generalisations, appear naturally as solutions of fractional differential equations. Furthermore, many interesting relationships between different special functions are found by using the operators of fractional calculus. Certain special functions have also been applied to analyse the qualitative properties of fractional differential equations, e.g., the concept of Mittag–Leffler stability.The aim of this reprint is to explore and highlight the diverse connections between fractional calculus and special functions, and their associated applications.

Research & information: general --- Mathematics & science --- Caputo-Hadamard fractional derivative --- coupled system --- Hadamard fractional integral --- boundary conditions --- existence --- fixed point theorem --- fractional Langevin equations --- existence and uniqueness solution --- fractional derivatives and integrals --- stochastic processes --- calculus of variations --- Mittag-Leffler functions --- Prabhakar fractional calculus --- Atangana-Baleanu fractional calculus --- complex integrals --- analytic continuation --- k-gamma function --- k-beta function --- Pochhammer symbol --- hypergeometric function --- Appell functions --- integral representation --- reduction and transformation formula --- fractional derivative --- generating function --- physical problems --- fractional derivatives --- fractional modeling --- real-world problems --- electrical circuits --- fractional differential equations --- fixed point theory --- Atangana-Baleanu derivative --- mobile phone worms --- fractional integrals --- Abel equations --- Laplace transforms --- mixed partial derivatives --- second Chebyshev wavelet --- system of Volterra-Fredholm integro-differential equations --- fractional-order Caputo derivative operator --- fractional-order Riemann-Liouville integral operator --- error bound --- Caputo-Hadamard fractional derivative --- coupled system --- Hadamard fractional integral --- boundary conditions --- existence --- fixed point theorem --- fractional Langevin equations --- existence and uniqueness solution --- fractional derivatives and integrals --- stochastic processes --- calculus of variations --- Mittag-Leffler functions --- Prabhakar fractional calculus --- Atangana-Baleanu fractional calculus --- complex integrals --- analytic continuation --- k-gamma function --- k-beta function --- Pochhammer symbol --- hypergeometric function --- Appell functions --- integral representation --- reduction and transformation formula --- fractional derivative --- generating function --- physical problems --- fractional derivatives --- fractional modeling --- real-world problems --- electrical circuits --- fractional differential equations --- fixed point theory --- Atangana-Baleanu derivative --- mobile phone worms --- fractional integrals --- Abel equations --- Laplace transforms --- mixed partial derivatives --- second Chebyshev wavelet --- system of Volterra-Fredholm integro-differential equations --- fractional-order Caputo derivative operator --- fractional-order Riemann-Liouville integral operator --- error bound

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This book describes the theory and applications of discrete orthogonal polynomials--polynomials that are orthogonal on a finite set. Unlike other books, Discrete Orthogonal Polynomials addresses completely general weight functions and presents a new methodology for handling the discrete weights case. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller focus on asymptotic aspects of general, nonclassical discrete orthogonal polynomials and set out applications of current interest. Topics covered include the probability theory of discrete orthogonal polynomial ensembles and the continuum limit of the Toda lattice. The primary concern throughout is the asymptotic behavior of discrete orthogonal polynomials for general, nonclassical measures, in the joint limit where the degree increases as some fraction of the total number of points of collocation. The book formulates the orthogonality conditions defining these polynomials as a kind of Riemann-Hilbert problem and then generalizes the steepest descent method for such a problem to carry out the necessary asymptotic analysis.

Orthogonal polynomials --- Asymptotic theory --- Orthogonal polynomials -- Asymptotic theory. --- Polynomials. --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Asymptotic theory. --- Asymptotic theory of orthogonal polynomials --- Algebra --- Airy function. --- Analytic continuation. --- Analytic function. --- Ansatz. --- Approximation error. --- Approximation theory. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Asymptotic formula. --- Beta function. --- Boundary value problem. --- Calculation. --- Cauchy's integral formula. --- Cauchy–Riemann equations. --- Change of variables. --- Complex number. --- Complex plane. --- Correlation function. --- Degeneracy (mathematics). --- Determinant. --- Diagram (category theory). --- Discrete measure. --- Distribution function. --- Eigenvalues and eigenvectors. --- Equation. --- Estimation. --- Existential quantification. --- Explicit formulae (L-function). --- Factorization. --- Fredholm determinant. --- Functional derivative. --- Gamma function. --- Gradient descent. --- Harmonic analysis. --- Hermitian matrix. --- Homotopy. --- Hypergeometric function. --- I0. --- Identity matrix. --- Inequality (mathematics). --- Integrable system. --- Invariant measure. --- Inverse scattering transform. --- Invertible matrix. --- Jacobi matrix. --- Joint probability distribution. --- Lagrange multiplier. --- Lax equivalence theorem. --- Limit (mathematics). --- Linear programming. --- Lipschitz continuity. --- Matrix function. --- Maxima and minima. --- Monic polynomial. --- Monotonic function. --- Morera's theorem. --- Neumann series. --- Number line. --- Orthogonal polynomials. --- Orthogonality. --- Orthogonalization. --- Parameter. --- Parametrix. --- Pauli matrices. --- Pointwise convergence. --- Pointwise. --- Polynomial. --- Potential theory. --- Probability distribution. --- Probability measure. --- Probability theory. --- Probability. --- Proportionality (mathematics). --- Quantity. --- Random matrix. --- Random variable. --- Rate of convergence. --- Rectangle. --- Rhombus. --- Riemann surface. --- Special case. --- Spectral theory. --- Statistic. --- Subset. --- Theorem. --- Toda lattice. --- Trace (linear algebra). --- Trace class. --- Transition point. --- Triangular matrix. --- Trigonometric functions. --- Uniform continuity. --- Unit vector. --- Upper and lower bounds. --- Upper half-plane. --- Variational inequality. --- Weak solution. --- Weight function. --- Wishart distribution. --- Orthogonal polynomials - Asymptotic theory

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This book develops arithmetic without the induction principle, working in theories that are interpretable in Raphael Robinson's theory Q. Certain inductive formulas, the bounded ones, are interpretable in Q. A mathematically strong, but logically very weak, predicative arithmetic is constructed.Originally published in 1986.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Constructive mathematics. --- Arithmetic. --- Mathematics --- Set theory --- Calculators --- Numbers, Real --- Mathematics, Constructive --- Logic, Symbolic and mathematical --- Addition. --- Adjunction (field theory). --- Age of the universe. --- Almost surely. --- Arithmetic IF. --- Atomic formula. --- Axiom. --- Axiomatic system. --- Beta function. --- Big O notation. --- Binary number. --- Binary relation. --- Brownian motion. --- Canonical form. --- Cardinality. --- Cartesian coordinate system. --- Chessboard. --- Classical mathematics. --- Closed-form expression. --- Commutative property. --- Computation. --- Conservative extension. --- Consistency. --- Contradiction. --- Deduction theorem. --- Diameter. --- Direct proof. --- Domain of discourse. --- Elementary mathematics. --- Elias M. Stein. --- Existential quantification. --- Exponential function. --- Exponentiation. --- Extension by definitions. --- Finitary. --- Finite set. --- Formula C (SCCA). --- Foundations of mathematics. --- Fundamenta Mathematicae. --- Gödel's completeness theorem. --- Herbrand's theorem. --- Impredicativity. --- Inaccessible cardinal. --- Inference. --- Interpretability. --- John Milnor. --- Logic. --- Logical connective. --- Mathematical induction. --- Mathematical logic. --- Mathematician. --- Mathematics. --- Measurable cardinal. --- Metamathematics. --- Metatheorem. --- Model theory. --- Mostowski. --- Natural number. --- Negation. --- Non-standard analysis. --- Notation. --- P-adic analysis. --- Peano axioms. --- Polynomial. --- Positional notation. --- Power of two. --- Power set. --- Primitive notion. --- Primitive recursive function. --- Principia Mathematica. --- Probability theory. --- Quantifier (logic). --- Quantity. --- Ranking (information retrieval). --- Rational number. --- Real number. --- Recursion (computer science). --- Remainder. --- Requirement. --- Robert Langlands. --- Rule of inference. --- Scientific notation. --- Sequence. --- Set theory. --- Subset. --- Theorem. --- Theory. --- Transfer principle. --- Transfinite number. --- Triviality (mathematics). --- Tuple. --- Uniqueness. --- Universal quantification. --- Variable (mathematics). --- Zermelo–Fraenkel set theory.

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This book focuses on applications of the theory of fractional calculus in numerical analysis and various fields of physics and engineering. Inequalities involving fractional calculus operators containing the Mittag–Leffler function in their kernels are of particular interest. Special attention is given to dynamical models, magnetization, hypergeometric series, initial and boundary value problems, and fractional differential equations, among others.

Research & information: general --- Mathematics & science --- fractional derivative --- generalized Mittag-Leffler kernel (GMLK) --- Legendre polynomials --- Legendre spectral collocation method --- dynamical systems --- random time change --- inverse subordinator --- asymptotic behavior --- Mittag–Leffler function --- data fitting --- magnetization --- magnetic fluids --- Gamma function --- Psi function --- Pochhammer symbol --- hypergeometric function 2F1 --- generalized hypergeometric functions tFu --- Gauss’s summation theorem for 2F1(1) --- Kummer’s summation theorem for 2F1(−1) --- generalized Kummer’s summation theorem for 2F1(−1) --- Stirling numbers of the first kind --- Hilfer–Hadamard fractional derivative --- Riemann–Liouville fractional derivative --- Caputo fractional derivative --- fractional differential equations --- inclusions --- nonlocal boundary conditions --- existence and uniqueness --- fixed point --- gamma function --- Beta function --- Mittag-Leffler function --- Generalized Mittag-Leffler functions --- generalized hypergeometric function --- Fox–Wright function --- recurrence relations --- Riemann–Liouville fractional calculus operators --- (α, h-m)-p-convex function --- Fejér–Hadamard inequality --- extended generalized fractional integrals --- Mittag–Leffler functions --- initial value problems --- Laplace transform --- exact solution --- Chebyshev inequality --- Pólya-Szegö inequality --- fractional integral operators --- Wright function --- Srivastava’s polynomials --- fractional calculus operators --- Lavoie–Trottier integral formula --- Oberhettinger integral formula --- fractional partial differential equation --- boundary value problem --- separation of variables --- Mittag-Leffler --- Abel-Gontscharoff Green’s function --- Hermite-Hadamard inequalities --- convex function --- κ-Riemann-Liouville fractional integral --- Dirichlet averages --- B-splines --- dirichlet splines --- Riemann–Liouville fractional integrals --- hypergeometric functions of one and several variables --- generalized Mittag-Leffler type function --- Srivastava–Daoust generalized Lauricella hypergeometric function --- fractional calculus --- Hermite–Hadamard inequality --- Fox H function --- subordinator and inverse stable subordinator --- Lamperti law --- order statistic --- n/a --- Gauss's summation theorem for 2F1(1) --- Kummer's summation theorem for 2F1(−1) --- generalized Kummer's summation theorem for 2F1(−1) --- Hilfer-Hadamard fractional derivative --- Riemann-Liouville fractional derivative --- Fox-Wright function --- Riemann-Liouville fractional calculus operators --- Fejér-Hadamard inequality --- Mittag-Leffler functions --- Pólya-Szegö inequality --- Srivastava's polynomials --- Lavoie-Trottier integral formula --- Abel-Gontscharoff Green's function --- Riemann-Liouville fractional integrals --- Srivastava-Daoust generalized Lauricella hypergeometric function --- Hermite-Hadamard inequality

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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