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Type of orthogonal polynomials
classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including
Classical orthogonal polynomials
Classical_orthogonal_polynomials
Set of polynomials where any two are orthogonal to each other
mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other
Orthogonal_polynomials
Classification of orthogonal polynomials
organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in
Askey_scheme
Family of orthogonal polynomials
include limiting cases of these polynomials, in which case it also includes the classical orthogonal polynomials. Hahn polynomials are defined in terms of generalized
Hahn_polynomials
Polynomial sequence
In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(α) n(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight
Gegenbauer_polynomials
Polynomial sequence
polynomials (occasionally called hypergeometric polynomials) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are a class of classical
Jacobi_polynomials
System of complete and orthogonal polynomials
mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of
Legendre_polynomials
Sequence of differential equation solutions
Zeros ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials". dlmf.nist.gov. "DLMF: §18.18 Sums ‣ Classical Orthogonal Polynomials ‣ Chapter
Laguerre_polynomials
Polynomial sequence
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets
Hermite_polynomials
Sobolev orthogonal polynomials in general no longer share some of the nice features that classical orthogonal polynomials have. Sobolev orthogonal polynomials
Sobolev orthogonal polynomials
Sobolev_orthogonal_polynomials
polynomials, Hermite-Padé polynomials or polyorthogonal polynomials. MOPs should not be confused with multivariate orthogonal polynomials. Consider a multiindex
Multiple orthogonal polynomials
Multiple_orthogonal_polynomials
Mathematics concept
for two other sets of orthogonal polynomials. In some contrast to the standard classical orthogonal polynomials, the polynomials under consideration differ
Romanovski_polynomials
Discrete orthogonal polynomials
polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname Кравчу́к) are discrete orthogonal polynomials
Kravchuk_polynomials
Orthogonal symmetric polynomial family
many other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials and Askey–Wilson polynomials, which in turn include
Macdonald_polynomials
In mathematics, Meixner polynomials, also called discrete Laguerre polynomials, are a family of discrete orthogonal polynomials introduced by Josef Meixner
Meixner_polynomials
Q-Hermite Polynomials and Classical Orthogonal Polynomials, arXiv:math/9405213 Al-Salam, W. A.; Carlitz, L. (1965), "Some orthogonal q-polynomials", Mathematische
Discrete q-Hermite polynomials
Discrete_q-Hermite_polynomials
Characterization theorem found in 1929
theory of orthogonal polynomials, Bochner's theorem is a characterization theorem of certain families of orthogonal polynomials as polynomial solutions
Bochner's theorem (orthogonal polynomials)
Bochner's_theorem_(orthogonal_polynomials)
Various meanings of the terms
families of functions are used to form an orthogonal basis, such as in the contexts of orthogonal polynomials, orthogonal functions, and combinatorics. In optics
Orthogonality
Polynomial sequence
In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike
Zernike_polynomials
Formula describing the asymptotic behavior of the Legendre polynomials
Legendre polynomials as the index tends to infinity, near the edges of the support of the weight. There are generalizations to other classical orthogonal polynomials
Mehler–Heine_formula
mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced
Koornwinder_polynomials
In mathematics, the q-Charlier polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter
Q-Charlier_polynomials
In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle
Orthogonal polynomials on the unit circle
Orthogonal_polynomials_on_the_unit_circle
mathematics, the continuous q-Legendre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Koekoek, Lesky
Continuous q-Legendre polynomials
Continuous_q-Legendre_polynomials
In mathematics, the q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme Roelof Koekoek, Peter
Q-Krawtchouk_polynomials
In mathematics, the q-Meixner polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter
Q-Meixner_polynomials
Family of orthogonal polynomials
continuous q-Jacobi polynomials P(α,β) n(x|q), introduced by Askey & Wilson (1985), are a family of basic hypergeometric orthogonal polynomials in the basic
Continuous q-Jacobi polynomials
Continuous_q-Jacobi_polynomials
Formula for the Legendre polynomials
it. The term is also used to describe similar formulas for other orthogonal polynomials. Askey (2005) describes the history of the Rodrigues formula in
Rodrigues'_formula
mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek
Continuous q-Hermite polynomials
Continuous_q-Hermite_polynomials
big q-Jacobi polynomials Pn(x;a,b,c;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. The polynomials are given in
Big_q-Jacobi_polynomials
Egyptian mathematician
of orthogonal polynomials and their linearization coefficients. Ismail, Mourad E. H. (2005), Classical and Quantum Orthogonal Polynomials in One Variable
Mourad_Ismail
Hermite polynomials Hermite polynomials, a sequence of polynomials orthogonal with respect to the normal distribution Continuous q-Hermite polynomials Continuous
List of things named after Charles Hermite
List_of_things_named_after_Charles_Hermite
Method of representing a random variable
random variable in terms of a polynomial function of other random variables. The polynomials are chosen to be orthogonal with respect to the joint probability
Polynomial_chaos
Topics referred to by the same term
linear algebra Hermite polynomials, a classical orthogonal polynomial sequence Hermite spline, a spline curve where each polynomial of the spline is in Hermite
Hermite_(disambiguation)
Generating polynomial of the number of ways to place non-attacking rooks on a chessboard
belongs to B. Thus, the theory of rook polynomials is, in a sense, contained in that of matching polynomials. We deduce an important fact about the coefficients
Rook_polynomial
French mathematician (1822–1901)
Mathematics portal List of things named after Charles Hermite Classical orthogonal polynomials G. B. H. (February 7, 1901). "Charles Hermite". Nature. 63
Charles_Hermite
Quasi-arithmetic mean Classical orthogonal polynomials Hermite polynomials Laguerre polynomials Jacobi polynomials Gegenbauer polynomials Legendre polynomials Euclidean
List_of_real_analysis_topics
In mathematics, a Wall polynomial is a polynomial studied by Wall in his work on conjugacy classes in classical groups, and named by George Andrews. Andrews
Wall_polynomial
little q-Laguerre polynomials pn(x;a|q) or Wall polynomials Wn(x; b,q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey
Little_q-Laguerre_polynomials
Concept in mathematics
cos(nx) are similar to the monomial basis for polynomials. In the complex case the trigonometric polynomials are spanned by the positive and negative powers
Trigonometric_polynomial
Family of power series in mathematics
functions as special cases, such as Bessel functions, and the classical orthogonal polynomials. A hypergeometric series is formally defined as a power series
Generalized hypergeometric function
Generalized_hypergeometric_function
Approximation of the definite integral of a function
well-approximated by polynomials on [ − 1 , 1 ] {\displaystyle [-1,1]} , the associated orthogonal polynomials are Legendre polynomials, denoted by Pn(x)
Gaussian_quadrature
Special mathematical functions defined on the surface of a sphere
harmonic polynomials R 3 → C that are homogeneous of degree ℓ } . {\displaystyle \mathbf {A} _{\ell }=\left\{{\text{harmonic polynomials }}\mathbb
Spherical_harmonics
Type of functions, in mathematical analysis
derangements. Hypergeometric functions, Bessel functions, and classical orthogonal polynomials, in addition to being holonomic functions of their variable
Holonomic_function
British mathematician
Verblunsky's theorem and Verblunsky coefficients. His early work on orthogonal polynomials and harmonic functions was neglected for many years, until publicized
Samuel_Verblunsky
Form of interpolation
polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials. The original use of interpolation polynomials was
Polynomial_interpolation
Hungarian mathematician
1954) is a Hungarian mathematician, working in classical analysis, harmonic analysis, orthogonal polynomials, approximation theory, potential theory. He
Vilmos_Totik
Number with an integer power equal to 1
coefficient in the nth cyclotomic polynomial. Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For
Root_of_unity
American mathematician (1933–2019)
organizes orthogonal polynomials of ( q {\displaystyle q} -)hypergeometric type into a hierarchy. The Askey–Gasper inequality for Jacobi polynomials is essential
Richard_Askey
Mathematical functions
Mittag-Leffler polynomials are the polynomials gn(x) or Mn(x) studied by Mittag-Leffler (1891). Mn(x) is a special case of the Meixner polynomial Mn(x;b,c)
Mittag-Leffler_polynomials
Generalization of beta distribution
ISBN 978-0-521-19452-5. Mehta, M.L. (2004). "19. Matrix ensembles and classical orthogonal polynomials". Random Matrices. Amsterdam: Elsevier/Academic Press. ISBN 0-12-088409-7
Matrix variate beta distribution
Matrix_variate_beta_distribution
Dunkl & Xu (2001), p. 308. Dunkl, Charles F.; Xu, Yuan (2001). Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and its Applications
Generalized_Pochhammer_symbol
Statistics concept
interval (0, 1). Although the correlation can be reduced by using orthogonal polynomials, it is generally more informative to consider the fitted regression
Polynomial_regression
Mathematical problem
combinatorics, two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Linear operator in mathematics
in the orthogonal polynomials. When these are orthogonal on the real number line, the shift is given by the Jacobi operator. When the polynomials are orthogonal
Composition_operator
British mathematician (1928–2023)
2. pp. 331–335. Macdonald, I. G. (1998). "Constant term polynomials, orthogonal polynomials, and affine Hecke algebras". Doc. Math. (Bielefeld) Extra
Ian_G._Macdonald
Group of unitary matrices
=-1} gives back the "classical" unitary group (as an algebraic group). The unitary groups are the automorphisms of two polynomials in real non-commutative
Unitary_group
the Gaussian matrix ensembles. The three main examples are the circular orthogonal ensemble (COE) on symmetric unitary matrices, the circular unitary ensemble
Circular_ensemble
Type of mathematical generalization
University Press, ISBN 0521833574. Ismail, M. E. H. (2005), Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press. Koekoek
Q-analog
Probability problem
see e.g. The Hamburger moment problem is intimately related to orthogonal polynomials on the real line. That is, assume { m n } n ∈ N 0 {\displaystyle
Hamburger_moment_problem
Model of logic based on matrix algebra
of logical operations as polynomials. For the case of monadic operators (such as identity or negation), the Boolean polynomials look as follows: f ( x )
Vector_logic
differintegral Generalized Fourier series Orthogonal functions Orthogonal polynomials Empirical orthogonal functions Set of uniqueness Continuous Fourier
List of harmonic analysis topics
List_of_harmonic_analysis_topics
Branch of mathematics studying functions of a complex variable
on Ω {\displaystyle \Omega } can be approximated arbitrarily well by polynomials in some neighborhood of every point in Ω {\displaystyle \Omega } . This
Complex_analysis
that it generates n + 1 orthogonal polynomials which can be used as orthonormal basis functions to expand any nth-order polynomial. It is closely related
Schur_class
American mathematician (1923-1995)
MR 0299761. Ullman, J. L. (1972). "On the regular behaviour of orthogonal polynomials" (PDF). Proc. London Math. Soc. 24 (3): 119–148. doi:10.1112/plms/s3-24
Joseph_L._Ullman
Orthonormalization of a set of vectors
often not quite orthogonal, due to rounding errors. For the Gram–Schmidt process as described above (sometimes referred to as "classical Gram–Schmidt")
Gram–Schmidt_process
formula that gives the characters of representations in terms of Schur polynomials, χ λ ( g ) = s λ ( x 1 , … , x n ) {\displaystyle \chi _{\lambda }(g)=s_{\lambda
Representations of classical groups
Representations_of_classical_groups
Mathematical problems related to differential equations
Riemann–Hilbert problems play a central role in integrable systems, orthogonal polynomials, random matrix theory, inverse monodromy, and asymptotic analysis
Riemann–Hilbert_problem
Geometry of the location of polynomial roots
real roots of a polynomial Root-finding of polynomials – Algorithms for finding zeros of polynomials Square-free polynomial – Polynomial with no repeated
Geometrical properties of polynomial roots
Geometrical_properties_of_polynomial_roots
Operator encoding information about iterated map
operator and the Hessenberg matrix, both of which generate systems of orthogonal polynomials via a right-shift. Whereas the iteration of a function f {\displaystyle
Transfer_operator
South African mathematician
T.-R.; Venakides, S.; Zhou, X. (1998). "Uniform asymptotic for orthogonal polynomials". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. III.
Percy_Deift
Subfield of computational complexity theory
in polynomial time. In 2004, Ryan Williams gave the reduction that connected SETH to fine-grained complexity. He showed that solving the Orthogonal Vectors
Fine_grained_complexity
Type of vector space in math
are frequently used to study orthogonal polynomials, because different families of orthogonal polynomials are orthogonal with respect to different weighting
Hilbert_space
Mathematical functions having established names and notations
and tabulation ceased to be the main issue. The modern theory of orthogonal polynomials is of a definite but limited scope. Hypergeometric series, observed
Special_functions
neighbouring entries to the top, topright and right. The Krawtchouk polynomials are orthogonal with respect to symmetric binomial distributions, p = 1 / 2 {\displaystyle
Krawtchouk_matrices
Random matrix with gaussian entries
Forrester, P. J.; Nagao, T.; van Moerbeke, P. (2000-04-01). "Classical Skew Orthogonal Polynomials and Random Matrices". Journal of Statistical Physics. 99
Gaussian_ensemble
American mathematician
fields, the semi-classical limit, the singular continuous spectrum, random and ergodic Schrödinger operators, orthogonal polynomials, and non-selfadjoint
Barry_Simon
Branch of mathematics
commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few
Algebraic_geometry
in classical analysis. His research mainly focuses on asymptotic analysis, singular perturbation theory, special functions and orthogonal polynomials, integral
Roderick_S._C._Wong
Study of mathematical knots
one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way.
Knot_theory
Topological quantum field theory
invariant polynomials from g (the Lie algebra of G) to the cohomology H ∗ ( M , R ) {\displaystyle H^{*}(M,\mathbb {R} )} . If the invariant polynomial is homogeneous
Chern–Simons_theory
case to general finite cyclic groups, the "classical phases"—essentially the exponentials of polynomials natural for the circle group—had proved inadequate
Nilsequence
Mathematical formula expressing equality
equation is a polynomial equation (commonly called also an algebraic equation) in which the two sides are polynomials. The sides of a polynomial equation contain
Equation
Process of developing trajectory performance
by a spline of a different order. The name comes from the use of orthogonal polynomials in the state and control splines. In pseudospectral discretization
Trajectory_optimization
Mathematical theorem
orthogonal polynomials[clarification needed] with respect to μ ∈ C, and let ξ1,...ξm be the zeros of Pm. It is not hard to see that the polynomials P0
Chebyshev–Markov–Stieltjes inequalities
Chebyshev–Markov–Stieltjes_inequalities
Array of numbers
+1 or −1. A special orthogonal matrix is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant
Matrix_(mathematics)
Group of 𝑛 × 𝑛 invertible matrices
and symmetries of vector spaces in general, as well as the study of polynomials. The modular group may be realised as a quotient of the special linear
General_linear_group
for the classical groups were determined by Weyl (1946) between successive unitary groups; Murnaghan (1938) between successive special orthogonal groups
Restricted_representation
Multi-dimensional generalization of triangle
being pairwise orthogonal to each other but not orthogonal to A 0 {\displaystyle A_{0}} , which is the facet opposite the orthogonal corner. For a 2-simplex
Simplex
Concepts from linear algebra
semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix
Eigenvalues_and_eigenvectors
Euclidean space Orthogonality Orthogonal complement Orthogonal projection Orthogonal group Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation
Outline_of_linear_algebra
a theorem about homogeneous polynomials. Euler polynomials Euler spline – splines composed of arcs using Euler polynomials Contributions of Leonhard Euler
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Group that is also a differentiable manifold with group operations that are smooth
showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early
Lie_group
Theory of getting acceptably close inexact mathematical calculations
a polynomial of degree N. One can obtain polynomials very close to the optimal one by expanding the given function in terms of Chebyshev polynomials and
Approximation_theory
embedding is complete; see rational normal curve. 2. Orthogonal to the tangent space, such as a line orthogonal to the tangent space or the normal bundle. 3. A
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
248-dimensional exceptional simple Lie group
large square matrices consisting of polynomials, the Lusztig–Vogan polynomials, an analogue of Kazhdan–Lusztig polynomials introduced for reductive groups
E8_(mathematics)
Branch of mathematics
various natures; for example, they could be tuples, sequences, functions, polynomials, or matrices. Linear algebra is concerned with the properties of such
Linear_algebra
Group of unitary complex matrices with determinant of 1
complex numbers with absolute value 1. For completeness, there are also the orthogonal and symplectic subgroups, SU ( n ) ⊃ SO ( n ) , SU ( 2 n ) ⊃ Sp
Special_unitary_group
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
minimal polynomial P of a square matrix A is the unique monic polynomial of least degree, m, such that P(A) = 0. Alternatively, the set of polynomials that
Jordan_normal_form
Branch of mathematics
the circle. It has the property of being an orthogonal expansion: any two of the eigenfunctions are orthogonal in the Hilbert space of square integrable
Mathematical_analysis
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CLASSICAL ORTHOGONAL-POLYNOMIALS
CLASSICAL ORTHOGONAL-POLYNOMIALS
CLASSICAL ORTHOGONAL-POLYNOMIALS
CLASSICAL ORTHOGONAL-POLYNOMIALS
CLASSICAL ORTHOGONAL-POLYNOMIALS
CLASSICAL ORTHOGONAL-POLYNOMIALS
CLASSICAL ORTHOGONAL-POLYNOMIALS
CLASSICAL ORTHOGONAL-POLYNOMIALS
CLASSICAL ORTHOGONAL-POLYNOMIALS
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