Orthogonal polynomials, the Glossary
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.[1]
Table of Contents
84 relations: Algebraic combinatorics, Andrey Markov, Appell sequence, Arthur Erdélyi, Askey scheme, Askey–Wilson polynomials, Binomial type, Biorthogonal polynomial, Charlier polynomials, Chebyshev polynomials, Classical orthogonal polynomials, Continued fraction, Continuous dual Hahn polynomials, Continuous Hahn polynomials, Discrete orthogonal polynomials, Dual Hahn polynomials, Electrostatics, Enumerative combinatorics, Favard's theorem, Gaussian quadrature, Gábor Szegő, Gegenbauer polynomials, Generalized Fourier series, Gram–Schmidt process, Hahn polynomials, Hall–Littlewood polynomials, Heckman–Opdam polynomials, Hermite polynomials, Inner product space, Integrable system, Jack function, Jacobi polynomials, Koornwinder polynomials, Kravchuk polynomials, Laguerre polynomials, Lévy process, Lebesgue–Stieltjes integration, Legendre polynomials, Lie group, Macdonald polynomials, Martingale (probability theory), Mathematical physics, Mathematics, Meixner polynomials, Meixner–Pollaczek polynomials, Moment (mathematics), Mourad Ismail, Naum Akhiezer, Number theory, Numerical analysis, ... Expand index (34 more) »
Algebraic combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
See Orthogonal polynomials and Algebraic combinatorics
Andrey Markov
Andrey Andreyevich Markov (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes.
See Orthogonal polynomials and Andrey Markov
Appell sequence
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence \_ satisfying the identity and in which p_0(x) is a non-zero constant.
See Orthogonal polynomials and Appell sequence
Arthur Erdélyi
Arthur Erdélyi FRS, FRSE (2 October 1908 – 12 December 1977) was a Hungarian-born British mathematician.
See Orthogonal polynomials and Arthur Erdélyi
Askey scheme
In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy.
See Orthogonal polynomials and Askey scheme
Askey–Wilson polynomials
In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by Richard Askey and James A. Wilson as q-analogs of the Wilson polynomials.
See Orthogonal polynomials and Askey–Wilson polynomials
Binomial type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers \left\ in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities Many such sequences exist.
See Orthogonal polynomials and Binomial type
Biorthogonal polynomial
In mathematics, a biorthogonal polynomial is a polynomial that is orthogonal to several different measures.
See Orthogonal polynomials and Biorthogonal polynomial
Charlier polynomials
In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier.
See Orthogonal polynomials and Charlier polynomials
Chebyshev polynomials
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x).
See Orthogonal polynomials and Chebyshev polynomials
Classical orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials).
See Orthogonal polynomials and Classical orthogonal polynomials
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.
See Orthogonal polynomials and Continued fraction
Continuous dual Hahn polynomials
In mathematics, the continuous dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials.
See Orthogonal polynomials and Continuous dual Hahn polynomials
Continuous Hahn polynomials
In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials.
See Orthogonal polynomials and Continuous Hahn polynomials
Discrete orthogonal polynomials
In mathematics, a sequence of discrete orthogonal polynomials is a sequence of polynomials that are pairwise orthogonal with respect to a discrete measure.
See Orthogonal polynomials and Discrete orthogonal polynomials
Dual Hahn polynomials
In mathematics, the dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials.
See Orthogonal polynomials and Dual Hahn polynomials
Electrostatics
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.
See Orthogonal polynomials and Electrostatics
Enumerative combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed.
See Orthogonal polynomials and Enumerative combinatorics
Favard's theorem
In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable 3-term recurrence relation is a sequence of orthogonal polynomials.
See Orthogonal polynomials and Favard's theorem
Gaussian quadrature
In numerical analysis, an -point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree or less by a suitable choice of the nodes and weights for.
See Orthogonal polynomials and Gaussian quadrature
Gábor Szegő
Gábor Szegő (January 20, 1895 – August 7, 1985) was a Hungarian-American mathematician.
See Orthogonal polynomials and Gábor Szegő
Gegenbauer polynomials
In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(x) are orthogonal polynomials on the interval with respect to the weight function (1 − x2)α–1/2.
See Orthogonal polynomials and Gegenbauer polynomials
Generalized Fourier series
In mathematics, a generalized Fourier series expands a square-integrable function defined on an interval over the real line.
See Orthogonal polynomials and Generalized Fourier series
Gram–Schmidt process
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other.
See Orthogonal polynomials and Gram–Schmidt process
Hahn polynomials
In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 and rediscovered by Wolfgang Hahn.
See Orthogonal polynomials and Hahn polynomials
Hall–Littlewood polynomials
In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ.
See Orthogonal polynomials and Hall–Littlewood polynomials
Heckman–Opdam polynomials
In mathematics, Heckman–Opdam polynomials (sometimes called Jacobi polynomials) Pλ(k) are orthogonal polynomials in several variables associated to root systems.
See Orthogonal polynomials and Heckman–Opdam polynomials
Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
See Orthogonal polynomials and Hermite polynomials
Inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product.
See Orthogonal polynomials and Inner product space
Integrable system
In mathematics, integrability is a property of certain dynamical systems.
See Orthogonal polynomials and Integrable system
Jack function
In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack.
See Orthogonal polynomials and Jack function
Jacobi polynomials
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of classical orthogonal polynomials.
See Orthogonal polynomials and Jacobi polynomials
Koornwinder polynomials
In mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced by Koornwinder and I. G. Macdonald, that generalize the Askey–Wilson polynomials.
See Orthogonal polynomials and Koornwinder polynomials
Kravchuk polynomials
Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname Кравчу́к) are discrete orthogonal polynomials associated with the binomial distribution, introduced by.
See Orthogonal polynomials and Kravchuk polynomials
Laguerre polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: xy + (1 - x)y' + ny.
See Orthogonal polynomials and Laguerre polynomials
Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions.
See Orthogonal polynomials and Lévy process
Lebesgue–Stieltjes integration
In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes both Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework.
See Orthogonal polynomials and Lebesgue–Stieltjes integration
Legendre polynomials
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications.
See Orthogonal polynomials and Legendre polynomials
Lie group
In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
See Orthogonal polynomials and Lie group
Macdonald polynomials
In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987.
See Orthogonal polynomials and Macdonald polynomials
Martingale (probability theory)
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values.
See Orthogonal polynomials and Martingale (probability theory)
Mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics.
See Orthogonal polynomials and Mathematical physics
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
See Orthogonal polynomials and Mathematics
Meixner polynomials
In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by.
See Orthogonal polynomials and Meixner polynomials
Meixner–Pollaczek polynomials
In mathematics, the Meixner–Pollaczek polynomials are a family of orthogonal polynomials P(x,φ) introduced by, which up to elementary changes of variables are the same as the Pollaczek polynomials P(x,a,b) rediscovered by in the case λ.
See Orthogonal polynomials and Meixner–Pollaczek polynomials
Moment (mathematics)
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph.
See Orthogonal polynomials and Moment (mathematics)
Mourad Ismail
Mourad E. H. Ismail (born April 27, 1944, in Cairo, Egypt) is a mathematician working on orthogonal polynomials and special functions.
See Orthogonal polynomials and Mourad Ismail
Naum Akhiezer
Naum Ilyich Akhiezer (Нау́м Іллі́ч Ахіє́зер; Нау́м Ильи́ч Ахие́зер; 6 March 1901 – 3 June 1980) was a Soviet and Ukrainian mathematician of Jewish origin, known for his works in approximation theory and the theory of differential and integral operators.
See Orthogonal polynomials and Naum Akhiezer
Number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions.
See Orthogonal polynomials and Number theory
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
See Orthogonal polynomials and Numerical analysis
Orthogonal polynomials on the unit circle
In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane, for some probability measure on the unit circle.
See Orthogonal polynomials and Orthogonal polynomials on the unit circle
Orthogonality
In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.
See Orthogonal polynomials and Orthogonality
Orthonormality
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors.
See Orthogonal polynomials and Orthonormality
Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev (p) (–) was a Russian mathematician and considered to be the founding father of Russian mathematics.
See Orthogonal polynomials and Pafnuty Chebyshev
Plancherel–Rotach asymptotics
The Plancherel–Rotach asymptotics are asymptotic results for orthogonal polynomials.
See Orthogonal polynomials and Plancherel–Rotach asymptotics
Polynomial
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.
See Orthogonal polynomials and Polynomial
Probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability.
See Orthogonal polynomials and Probability theory
Q-analog
In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as.
See Orthogonal polynomials and Q-analog
Quantum group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure.
See Orthogonal polynomials and Quantum group
Racah polynomials
In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients.
See Orthogonal polynomials and Racah polynomials
Random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution.
See Orthogonal polynomials and Random matrix
Rehuel Lobatto
Rehuel Lobatto (6 June 1797 – 9 February 1866) was a Dutch mathematician.
See Orthogonal polynomials and Rehuel Lobatto
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.
See Orthogonal polynomials and Representation theory
Richard Askey
Richard Allen Askey (June 4, 1933 – October 9, 2019) was an American mathematician, known for his expertise in the area of special functions.
See Orthogonal polynomials and Richard Askey
Rogers–Szegő polynomials
In mathematics, the Rogers–Szegő polynomials are a family of polynomials orthogonal on the unit circle introduced by, who was inspired by the continuous q-Hermite polynomials studied by Leonard James Rogers.
See Orthogonal polynomials and Rogers–Szegő polynomials
Secondary measure
In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system.
See Orthogonal polynomials and Secondary measure
Sergei Bernstein
Sergei Natanovich Bernstein (Сергі́й Ната́нович Бернште́йн, sometimes Romanized as Bernshtein; 5 March 1880 – 26 October 1968) was a Ukrainian and Russian mathematician of Jewish origin known for contributions to partial differential equations, differential geometry, probability theory, and approximation theory.
See Orthogonal polynomials and Sergei Bernstein
Sheffer sequence
In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics.
See Orthogonal polynomials and Sheffer sequence
Sieved Jacobi polynomials
In mathematics, sieved Jacobi polynomials are a family of sieved orthogonal polynomials, introduced by.
See Orthogonal polynomials and Sieved Jacobi polynomials
Sieved orthogonal polynomials
In mathematics, sieved orthogonal polynomials are orthogonal polynomials whose recurrence relations are formed by sieving the recurrence relations of another family; in other words, some of the recurrence relations are replaced by simpler ones.
See Orthogonal polynomials and Sieved orthogonal polynomials
Sieved Pollaczek polynomials
In mathematics, sieved Pollaczek polynomials are a family of sieved orthogonal polynomials, introduced by.
See Orthogonal polynomials and Sieved Pollaczek polynomials
Sieved ultraspherical polynomials
In mathematics, the two families c(x;k) and B(x;k) of sieved ultraspherical polynomials, introduced by Waleed Al-Salam, W.R. Allaway and Richard Askey in 1984, are the archetypal examples of sieved orthogonal polynomials.
See Orthogonal polynomials and Sieved ultraspherical polynomials
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order.
See Orthogonal polynomials and Sobolev space
Sturm–Liouville theory
In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form for given functions p(x), q(x) and w(x), together with some boundary conditions at extreme values of x. The goals of a given Sturm–Liouville problem are.
See Orthogonal polynomials and Sturm–Liouville theory
The Mathematical Intelligencer
The Mathematical Intelligencer is a mathematical journal published by Springer Science+Business Media that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals.
See Orthogonal polynomials and The Mathematical Intelligencer
Theodore Seio Chihara
Theodore Seio Chihara (born 1929) is an American mathematician working on orthogonal polynomials who introduced Al-Salam–Chihara polynomials, Brenke–Chihara polynomials, and Chihara–Ismail polynomials.
See Orthogonal polynomials and Theodore Seio Chihara
Thomas Joannes Stieltjes
Thomas Joannes Stieltjes (29 December 1856 – 31 December 1894) was a Dutch mathematician.
See Orthogonal polynomials and Thomas Joannes Stieltjes
Umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to "prove" them.
See Orthogonal polynomials and Umbral calculus
Vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''.
See Orthogonal polynomials and Vector space
Waleed Al-Salam
Waleed Al-Salam (born 15 July 1926 in Baghdad, Iraq – died 14 April 1996 in Edmonton, Canada) was a mathematician who introduced Al-Salam–Chihara polynomials, Al-Salam–Carlitz polynomials, q-Konhauser polynomials, and Al-Salam–Ismail polynomials.
See Orthogonal polynomials and Waleed Al-Salam
Wilson polynomials
In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials.
See Orthogonal polynomials and Wilson polynomials
Wolfgang Hahn
Wolfgang Hahn (April 30, 1911 – January 10, 1998) was a German mathematician who worked on special functions, in particular orthogonal polynomials.
See Orthogonal polynomials and Wolfgang Hahn
Yakov Geronimus
Yakov Lazarevich Geronimus, sometimes spelled J. Geronimus (Я́ков Лазаре́вич Геро́нимус; February 6, 1898, Rostov – July 17, 1984, Kharkov) was a Russian mathematician known for contributions to theoretical mechanics and the study of orthogonal polynomials.
See Orthogonal polynomials and Yakov Geronimus
Zernike polynomials
In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk.
See Orthogonal polynomials and Zernike polynomials
References
[1] https://en.wikipedia.org/wiki/Orthogonal_polynomials
Also known as Orthogonal polynomial, Orthogonal polynomials/Proofs, Orthonormal polynomial.
, Orthogonal polynomials on the unit circle, Orthogonality, Orthonormality, Pafnuty Chebyshev, Plancherel–Rotach asymptotics, Polynomial, Probability theory, Q-analog, Quantum group, Racah polynomials, Random matrix, Rehuel Lobatto, Representation theory, Richard Askey, Rogers–Szegő polynomials, Secondary measure, Sergei Bernstein, Sheffer sequence, Sieved Jacobi polynomials, Sieved orthogonal polynomials, Sieved Pollaczek polynomials, Sieved ultraspherical polynomials, Sobolev space, Sturm–Liouville theory, The Mathematical Intelligencer, Theodore Seio Chihara, Thomas Joannes Stieltjes, Umbral calculus, Vector space, Waleed Al-Salam, Wilson polynomials, Wolfgang Hahn, Yakov Geronimus, Zernike polynomials.