01/2019, ISBN 9783039216215

.... Each paper presents mathematical theories, methods, and their application based on current and recently developed symmetric polynomials...

eBook

Journal of Mathematical Physics, ISSN 1089-7658, 2011, Volume 52, Issue 10, pp. 102301 - 102301-52

... generated as real representations by Mellin transforms of Poincaré–iterated integrals, including denominators of higher cyclotomic polynomials...

ANALYTIC CONTINUATION | polynomials | NUMERICAL EVALUATION | PHYSICS, MATHEMATICAL | ALGEBRAIC RELATIONS | HYPERGEOMETRIC-FUNCTIONS | integral equations | harmonics | perturbation theory | ONE-LOOP | renormalisation | MELLIN TRANSFORMS | LEGENDRE CHI | Feynman diagrams | MULTIPLE ZETA VALUES | NESTED SUMS

ANALYTIC CONTINUATION | polynomials | NUMERICAL EVALUATION | PHYSICS, MATHEMATICAL | ALGEBRAIC RELATIONS | HYPERGEOMETRIC-FUNCTIONS | integral equations | harmonics | perturbation theory | ONE-LOOP | renormalisation | MELLIN TRANSFORMS | LEGENDRE CHI | Feynman diagrams | MULTIPLE ZETA VALUES | NESTED SUMS

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 10/2017, Volume 322, pp. 25 - 45

.... To obtain highly accurate result for European call option, the implementation involves integrating high degree Legendre polynomials against exponential function...

Characteristic function | European option pricing | Legendre polynomials | Fourier series | Olver algorithm | MATHEMATICS, APPLIED | SERIES | Analysis | Pricing | Algorithms

Characteristic function | European option pricing | Legendre polynomials | Fourier series | Olver algorithm | MATHEMATICS, APPLIED | SERIES | Analysis | Pricing | Algorithms

Journal Article

Proceedings of the American Mathematical Society, ISSN 0002-9939, 06/2011, Volume 139, Issue 6, pp. 1915 - 1929

Let p be an odd prime. In this paper, by using the properties of Legendre polynomials we prove some congruences for $\sum\nolimits_{k = 0}^{\frac{{p - 1}}{2}} {{{(_k^{2k})}^2}{m^{ - k}}}$ (m odp₂...

Integers | Mathematical congruence | Legendre polynomials | Number theory | Legendre polynomial | Congruence | MATHEMATICS | MATHEMATICS, APPLIED | NUMBER | HYPERGEOMETRIC-SERIES | SUPERCONGRUENCES | congruence | Mathematics - Number Theory

Integers | Mathematical congruence | Legendre polynomials | Number theory | Legendre polynomial | Congruence | MATHEMATICS | MATHEMATICS, APPLIED | NUMBER | HYPERGEOMETRIC-SERIES | SUPERCONGRUENCES | congruence | Mathematics - Number Theory

Journal Article

Annals of physics, ISSN 0003-4916, 2013, Volume 335, pp. 78 - 85

We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects...

Group representation theory | Quantum mechanics | Coherent states | Orthogonal polynomials | PHYSICS, MULTIDISCIPLINARY | Algebra | Polynomials | Lie groups | Hilbert space | Coherence | Differential equations | DIFFERENTIAL EQUATIONS | INTEGRAL CALCULUS | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | ALGEBRA | EIGENSTATES | MATRIX ELEMENTS | QUANTUM MECHANICS | HILBERT SPACE | LEGENDRE POLYNOMIALS | RECURSION RELATIONS | HERMITE POLYNOMIALS | ANNIHILATION OPERATORS | LIE GROUPS

Group representation theory | Quantum mechanics | Coherent states | Orthogonal polynomials | PHYSICS, MULTIDISCIPLINARY | Algebra | Polynomials | Lie groups | Hilbert space | Coherence | Differential equations | DIFFERENTIAL EQUATIONS | INTEGRAL CALCULUS | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | ALGEBRA | EIGENSTATES | MATRIX ELEMENTS | QUANTUM MECHANICS | HILBERT SPACE | LEGENDRE POLYNOMIALS | RECURSION RELATIONS | HERMITE POLYNOMIALS | ANNIHILATION OPERATORS | LIE GROUPS

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 05/2015, Volume 84, Issue 293, pp. 1317 - 1337

Over the past few years considerable attention has been given to the role played by the Zernike polynomials (ZPs...

Quaternionic analysis | Spherical aberrations | Zernike polynomials | Spherical monogenics | Ferrer's associated legendre functions | Chebyshev polynomials | MATHEMATICS, APPLIED | spherical aberrations | Ferrer's associated Legendre functions | spherical monogenics

Quaternionic analysis | Spherical aberrations | Zernike polynomials | Spherical monogenics | Ferrer's associated legendre functions | Chebyshev polynomials | MATHEMATICS, APPLIED | spherical aberrations | Ferrer's associated Legendre functions | spherical monogenics

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 09/2019, Volume 88, Issue 319, pp. 2351 - 2381

We present a general framework for calculating the Volterra-type convolution of polynomials from an arbitrary polynomial sequence \{P_k(x)\}_{k \geqslant 0} with \deg P_k(x) = k...

Jacobi polynomials | MATHEMATICS, APPLIED | Gegenbauer polynomials | Convolution | Laguerre polynomials | REPRESENTATIONS | Volterra convolution integral | CONNECTION COEFFICIENTS | Legendre polynomials | Chebyshev polynomials | orthogonal polynomials

Jacobi polynomials | MATHEMATICS, APPLIED | Gegenbauer polynomials | Convolution | Laguerre polynomials | REPRESENTATIONS | Volterra convolution integral | CONNECTION COEFFICIENTS | Legendre polynomials | Chebyshev polynomials | orthogonal polynomials

Journal Article

1946, 42

Book

Applicable Analysis and Discrete Mathematics, ISSN 1452-8630, 10/2018, Volume 12, Issue 2, pp. 362 - 388

Using the techniques of the modern umbral calculus, we derive several combinatorial identities involving -Appell polynomials...

Integers | Numbers | Mathematical theorems | Umbral calculus | Discrete mathematics | Polynomials | Hermite polynomials | Binomials | Legendre polynomials | Combinatorial sums | Formal series | Generating functions | Orthogonal polynomials | MATHEMATICS | MATHEMATICS, APPLIED | NUMBERS | umbral calculus | formal series | generating functions | orthogonal polynomials

Integers | Numbers | Mathematical theorems | Umbral calculus | Discrete mathematics | Polynomials | Hermite polynomials | Binomials | Legendre polynomials | Combinatorial sums | Formal series | Generating functions | Orthogonal polynomials | MATHEMATICS | MATHEMATICS, APPLIED | NUMBERS | umbral calculus | formal series | generating functions | orthogonal polynomials

Journal Article

SIAM journal on scientific computing, ISSN 1095-7197, 2018, Volume 40, Issue 4, pp. A2336 - A2355

We develop a unified framework for constructing matrix approximations to the convolution operator of Volterra type defined by functions that are approximated using classical orthogonal polynomials on [-1,1...

Jacobi polynomials | Operator approximation | Gegenbauer polynomials | Convolution | Laguerre polynomials | Volterra convolution integral | Orthogonal polynomials | Legendre polynomials | Ultraspherical polynomials | Chebyshev polynomials | Spectral methods | MATHEMATICS, APPLIED | spectral methods | ALGORITHM | ultraspherical polynomials | convolution | operator approximation | orthogonal polynomials

Jacobi polynomials | Operator approximation | Gegenbauer polynomials | Convolution | Laguerre polynomials | Volterra convolution integral | Orthogonal polynomials | Legendre polynomials | Ultraspherical polynomials | Chebyshev polynomials | Spectral methods | MATHEMATICS, APPLIED | spectral methods | ALGORITHM | ultraspherical polynomials | convolution | operator approximation | orthogonal polynomials

Journal Article

SIAM Journal on Mathematical Analysis, ISSN 0036-1410, 2017, Volume 49, Issue 1, pp. 626 - 645

SIAM J. Math. Anal. 49 (2017), no. 1, 626-645 We show that the multiwavelets, introduced by Alpert in 1993, are related to type I Legendre-Angelesco multiple orthogonal polynomials...

Hypergeometric functions | Alpert multiwavelets | Legendre polynomials | Legendre-Angelesco polynomials | Mathematics - Classical Analysis and ODEs

Hypergeometric functions | Alpert multiwavelets | Legendre polynomials | Legendre-Angelesco polynomials | Mathematics - Classical Analysis and ODEs

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 11/2014, Volume 83, Issue 290, pp. 2893 - 2914

Barycentric interpolation is arguably the method of choice for numerical polynomial interpolation...

Hypergeometric functions | Interpolation | Roots of functions | Approximation | Polynomials | Hermite polynomials | Legendre polynomials | Gaussian quadratures | Weighting functions | Degrees of polynomials | JACOBI | NODES | MATHEMATICS, APPLIED | GAUSS-LEGENDRE | APPROXIMATION | STABILITY | DIFFERENTIAL-EQUATIONS | COMPUTATION | FORMULAS | LAGRANGE INTERPOLATION

Hypergeometric functions | Interpolation | Roots of functions | Approximation | Polynomials | Hermite polynomials | Legendre polynomials | Gaussian quadratures | Weighting functions | Degrees of polynomials | JACOBI | NODES | MATHEMATICS, APPLIED | GAUSS-LEGENDRE | APPROXIMATION | STABILITY | DIFFERENTIAL-EQUATIONS | COMPUTATION | FORMULAS | LAGRANGE INTERPOLATION

Journal Article

Journal of Applied Mathematics and Computing, ISSN 1598-5865, 10/2018, Volume 58, Issue 1, pp. 75 - 94

... problems by Müntz–Legendre polynomials Y. Ordokhani 1 · P. Rahimkhani 1,2 Received: 29 July 2017 / Published online: 11 September 2017 © Korean Society...

Computational Mathematics and Numerical Analysis | Fractional variational problems | Rayleigh–Ritz method | Mathematics | Theory of Computation | Numerical method | Caputo fractional derivative | 34A08 | Mathematics of Computing | Mathematical and Computational Engineering | 34K28 | Müntz–Legendre polynomials | 65L10 | Rayleigh-Ritz method | MATHEMATICS | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | BERNOULLI WAVELETS | Muntz-Legendre polynomials | DIFFERENTIAL-EQUATIONS | FORMULATION | EULER-LAGRANGE EQUATIONS | Norms | Error detection | Ritz method | Iterative methods | Mathematical analysis | Fractional calculus

Computational Mathematics and Numerical Analysis | Fractional variational problems | Rayleigh–Ritz method | Mathematics | Theory of Computation | Numerical method | Caputo fractional derivative | 34A08 | Mathematics of Computing | Mathematical and Computational Engineering | 34K28 | Müntz–Legendre polynomials | 65L10 | Rayleigh-Ritz method | MATHEMATICS | OPERATIONAL MATRIX | MATHEMATICS, APPLIED | BERNOULLI WAVELETS | Muntz-Legendre polynomials | DIFFERENTIAL-EQUATIONS | FORMULATION | EULER-LAGRANGE EQUATIONS | Norms | Error detection | Ritz method | Iterative methods | Mathematical analysis | Fractional calculus

Journal Article

Journal of number theory, ISSN 0022-314X, 2013, Volume 133, Issue 6, pp. 1950 - 1976

...). In particular, we show that ∑k=0p−12(2kk)3≡0(modp2) for p≡3,5,6(mod7). Let {Pn(x)} be the Legendre polynomials...

Binary quadratic form | Congruence | Character sum | Elliptic curve | Legendre polynomial | Legendre p-lynomial | Ellip-ic curve | ELLIPTIC-CURVES | BINOMIAL COEFFICIENTS | CHARACTER SUMS | HYPERGEOMETRIC-SERIES | MATHEMATICS | APERY NUMBERS | SUPERCONGRUENCES | COMPLEX MULTIPLICATION

Binary quadratic form | Congruence | Character sum | Elliptic curve | Legendre polynomial | Legendre p-lynomial | Ellip-ic curve | ELLIPTIC-CURVES | BINOMIAL COEFFICIENTS | CHARACTER SUMS | HYPERGEOMETRIC-SERIES | MATHEMATICS | APERY NUMBERS | SUPERCONGRUENCES | COMPLEX MULTIPLICATION

Journal Article

Advances in difference equations, ISSN 1687-1847, 2019, Volume 2019, Issue 1, pp. 1 - 16

The classical linearization problem concerns with determining the coefficients in the expansion of the product of two polynomials in terms of any given sequence of polynomials...

Hermite polynomial | Sums of finite products | Jacobi polynomial | Extended Laguerre polynomial | Mathematics | Legendre polynomial | Ordinary Differential Equations | Functional Analysis | Analysis | Difference and Functional Equations | Mathematics, general | Partial Differential Equations | Chebyshev polynomials | Gegenbauer polynomial | MATHEMATICS | MATHEMATICS, APPLIED | Hypergeometric functions | Thermal expansion | Chebyshev approximation | Polynomials | Representations | Linearization | Sums

Hermite polynomial | Sums of finite products | Jacobi polynomial | Extended Laguerre polynomial | Mathematics | Legendre polynomial | Ordinary Differential Equations | Functional Analysis | Analysis | Difference and Functional Equations | Mathematics, general | Partial Differential Equations | Chebyshev polynomials | Gegenbauer polynomial | MATHEMATICS | MATHEMATICS, APPLIED | Hypergeometric functions | Thermal expansion | Chebyshev approximation | Polynomials | Representations | Linearization | Sums

Journal Article

Computer methods in applied mechanics and engineering, ISSN 0045-7825, 2015, Volume 290, Issue C, pp. 73 - 97

Independent sampling of orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models using Polynomial Chaos (PC) expansions...

Uncertainty quantification | Hermite polynomials | Legendre polynomials | Markov chain Monte Carlo | Least squares regression | Polynomial Chaos | Markov chain monte carlo | Polynomial chaos | EXPANSIONS | STOCHASTIC COLLOCATION | PROJECTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | PARTIAL-DIFFERENTIAL-EQUATIONS | Monte Carlo method | Analysis | Markov processes | Differential equations | Aerospace engineering | Statistical analysis | Samples | Coherence | Mathematical models | Polynomials | Statistical methods | Sampling | Personal computers

Uncertainty quantification | Hermite polynomials | Legendre polynomials | Markov chain Monte Carlo | Least squares regression | Polynomial Chaos | Markov chain monte carlo | Polynomial chaos | EXPANSIONS | STOCHASTIC COLLOCATION | PROJECTION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | PARTIAL-DIFFERENTIAL-EQUATIONS | Monte Carlo method | Analysis | Markov processes | Differential equations | Aerospace engineering | Statistical analysis | Samples | Coherence | Mathematical models | Polynomials | Statistical methods | Sampling | Personal computers

Journal Article

Electronic Journal of Combinatorics, ISSN 1077-8926, 01/2017, Volume 24, Issue 1

.... Kontsevich i i cve conjectured in 1997 that the number of nonzero values of s(alpha, G) is a polynomial in q for all graphs...

Feynman amplitudes | Laplacian matrix | Flow polynomial | Kontsevich’s conjecture | Tutte 5-flow conjecture | Legendre symbol | MATHEMATICS | MATHEMATICS, APPLIED | flow polynomial | Kontsevich's conjecture | CONJECTURE

Feynman amplitudes | Laplacian matrix | Flow polynomial | Kontsevich’s conjecture | Tutte 5-flow conjecture | Legendre symbol | MATHEMATICS | MATHEMATICS, APPLIED | flow polynomial | Kontsevich's conjecture | CONJECTURE

Journal Article

Journal of Approximation Theory, ISSN 0021-9045, 01/2018, Volume 225, pp. 242 - 283

Polynomial perturbations of real multivariate measures are discussed and corresponding Christoffel type formulas are found...

Darboux transformations | Christoffel transformations | Bivariate product Legendre polynomials | Multivariate orthogonal polynomials | Algebraic varieties | Borel–Gauss factorization | MATHEMATICS | DETERMINANTS | Borel-Gauss factorization | LINEAR-EQUATIONS

Darboux transformations | Christoffel transformations | Bivariate product Legendre polynomials | Multivariate orthogonal polynomials | Algebraic varieties | Borel–Gauss factorization | MATHEMATICS | DETERMINANTS | Borel-Gauss factorization | LINEAR-EQUATIONS

Journal Article

Journal of number theory, ISSN 0022-314X, 2014, Volume 143, pp. 293 - 319

For any positive integer n and variables a and x we define the generalized Legendre polynomial Pn(a,x) by Pn(a,x)=∑k=0n(ak)(−1−ak)(1−x2)k...

Generalized Legendre polynomial | Binomial coefficient | Congruence | MATHEMATICS | PRODUCTS | BINOMIAL COEFFICIENTS | SUMS

Generalized Legendre polynomial | Binomial coefficient | Congruence | MATHEMATICS | PRODUCTS | BINOMIAL COEFFICIENTS | SUMS

Journal Article

Tamkang Journal of Mathematics, ISSN 0049-2930, 06/2015, Volume 46, Issue 2, pp. 167 - 177

Journal Article

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