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Bessel polynomials

In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by

Szegő polynomial

In mathematics, a Szegő polynomial is one of a family of orthogonal polynomials for the Hermitian inner product where dμ is a given positive measure on [−π, π]. Writing for the polynomials, they obey

Charlier polynomials

In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier.They are given in terms of the generalized hypergeome

Chebyshev polynomials

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonome

Bender–Dunne polynomials

In mathematics, Bender–Dunne polynomials are a two-parameter family of sequences of orthogonal polynomials studied by Carl M. Bender and Gerald Dunne . They may be defined by the recursion: , , and fo

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. They are defined in terms of generalized hyper

Pseudo-Zernike polynomials

In mathematics, pseudo-Zernike polynomials are well known and widely used in the analysis of optical systems. They are also widely used in image analysis as shape descriptors.

Big q-Jacobi polynomials

In mathematics, the big q-Jacobi polynomials Pn(x;a,b,c;q), introduced by , are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and R

Dual Hahn polynomials

In mathematics, the dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined on a non-uniform lattice and are defined

Brenke–Chihara polynomials

In mathematics, Brenke polynomials are special cases of generalized Appell polynomials, and Brenke–Chihara polynomials are the Brenke polynomials that are also orthogonal polynomials. Brenke introduce

Hermite polynomials

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in:
* signal processing as Hermitian wavelets for wavelet transform analysis
* probabili

Little q-Laguerre polynomials

In mathematics, the 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 scheme closely related to a co

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 Wo

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 me

Askey–Wilson polynomials

In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by Askey and Wilson as q-analogs of the Wilson polynomials. They include many o

Gottlieb polynomials

In mathematics, Gottlieb polynomials are a family of discrete orthogonal polynomials introduced by Morris J. Gottlieb. They are given by

Mehler kernel

The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator.

Continuous q-Laguerre polynomials

In mathematics, the continuous q-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give

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 di

Al-Salam–Chihara polynomials

In mathematics, the Al-Salam–Chihara polynomials Qn(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Al-Salam and Chihara. Roelof Koekoek,

Big q-Legendre polynomials

In mathematics, the big q-Legendre polynomials are an orthogonal family of polynomials defined in terms of Heine's basic hypergeometric series as . They obey the orthogonality relation and have the li

Dual q-Krawtchouk polynomials

In mathematics, the dual q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give a de

Wilson polynomials

In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by James A. Wilsonthat generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined

Tricomi–Carlitz polynomials

In mathematics, the Tricomi–Carlitz polynomials or (Carlitz–)Karlin–McGregor polynomials are polynomials studied by Tricomi and Carlitz and Karlin and McGregor, related to random walks on the positive

Big q-Laguerre polynomials

In mathematics, the big q-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give a detai

Pseudo Jacobi polynomials

In mathematics, the term Pseudo Jacobi polynomials was introduced by Lesky for one of three finite sequences of orthogonal polynomials y. Since they form an orthogonal subset of Routh polynomials it s

Rodrigues' formula

In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by Olinde Rodrigues, Sir James Ivory and Carl Gustav Ja

Gegenbauer polynomials

In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(α)n(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − x2)α–1/2. They generalize Legen

Discrete q-Hermite polynomials

In mathematics, the discrete q-Hermite polynomials are two closely related families hn(x;q) and ĥn(x;q) of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Al-Salam

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.Examples include the discrete Chebyshev polyn

Hall–Littlewood polynomials

In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ. They are Schur functions when t is 0 and monomial symmetric functions when t is 1

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.

Q-Meixner polynomials

In mathematics, the q-Meixner polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give a detailed l

Q-Racah polynomials

In mathematics, the q-Racah polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw giv

Q-Charlier polynomials

In mathematics, the q-Charlier polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give a detailed

Legendre polynomials

In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of ma

Al-Salam–Carlitz polynomials

In mathematics, Al-Salam–Carlitz polynomials U(a)n(x;q) and V(a)n(x;q) are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Waleed Al-Salam and Leon

Chihara–Ismail polynomials

In mathematics, the Chihara–Ismail polynomials are a family of orthogonal polynomials introduced by Chihara and Ismail, generalizing the van Doorn polynomials introduced by and the Karlin–McGregor pol

Jack function

In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal

Quantum q-Krawtchouk polynomials

In mathematics, the quantum q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give a

Zernike polynomials

In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, winner of the 1953 Nobel Prize in Physics and t

Biorthogonal polynomial

In mathematics, a biorthogonal polynomial is a polynomial that is orthogonal to several different measures. Biorthogonal polynomials are a generalization of orthogonal polynomials and share many of th

Continuous big q-Hermite polynomials

In mathematics, the continuous big q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw gi

Affine root system

In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebras and superalgebras, and semisimple p-a

Sieved Jacobi polynomials

In mathematics, sieved Jacobi polynomials are a family of sieved orthogonal polynomials, introduced by . Their recurrence relations are a modified (or "sieved") version of the recurrence relations for

Konhauser polynomials

In mathematics, the Konhauser polynomials, introduced by Konhauser, are biorthogonal polynomials for the distribution function of the Laguerre polynomials.

Askey–Gasper inequality

In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey and George Gasper and used in the proof of the Bieberbach conjecture.

Mehler–Heine formula

In mathematics, the Mehler–Heine formula introduced by Gustav Ferdinand Mehler and Eduard Heine describes the asymptotic behavior of the Legendre polynomials as the index tends to infinity, near the e

Rogers polynomials

In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by Rogers in the cour

Romanovski polynomials

In mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski in French transcription) within the context of p

Q-Laguerre polynomials

In mathematics, the q-Laguerre polynomials, or generalized Stieltjes–Wigert polynomials P(α)n(x;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by D

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. He later introduced a non-symmetric generalizat

N! conjecture

In mathematics, the n! conjecture is the conjecture that the dimension of a certain module of is n!. It was made by A. M. Garsia and M. Haiman and later proved by M. Haiman. It implies Macdonald's pos

Q-Konhauser polynomials

In mathematics, the q-Konhauser polynomials are a q-analog of the Konhauser polynomials, introduced by .

Rook polynomial

In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in

Legendre moment

In mathematics, Legendre moments are a type of image moment and are achieved by using the Legendre polynomial. Legendre moments are used in areas of image processing including: pattern and object reco

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 (1987

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. The Racah

Askey scheme

In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in ,

Continuous q-Legendre polynomials

In mathematics, the continuous q-Legendre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.

Affine q-Krawtchouk polynomials

In mathematics, the affine q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Carlitz and Hodges. Roelof Koekoek, Peter A. Le

Sieved ultraspherical polynomials

In mathematics, the two families cλn(x;k) and Bλn(x;k) of sieved ultraspherical polynomials, introduced by Waleed Al-Salam, W.R. Allaway and Richard Askey in 1984, are the archetypal examples of sieve

Q-Bessel polynomials

In mathematics, the q-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give a detailed li

Q-Meixner–Pollaczek polynomials

In mathematics, the q-Meixner–Pollaczek polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give a

Dual q-Hahn polynomials

In mathematics, the dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give a detailed

Laguerre polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: which is a second-order linear differential equation. This equation has nonsing

Continuous dual q-Hahn polynomials

In mathematics, the continuous dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give

Schur polynomial

In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the comple

Continuous q-Hermite polynomials

In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give a

Continuous q-Jacobi polynomials

In mathematics, the continuous q-Jacobi polynomials P(α,β)n(x|q), introduced by , are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky,

Stieltjes polynomials

In mathematics, the Stieltjes polynomials En are polynomials associated to a family of orthogonal polynomials Pn. They are unrelated to the Stieltjes polynomial solutions of differential equations. St

Rogers–Szegő polynomials

In mathematics, the Rogers–Szegő polynomials are a family of polynomials orthogonal on the unit circle introduced by Szegő, who was inspired by the continuous q-Hermite polynomials studied by Leonard

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

Meixner polynomials

In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by Josef Meixner. They are given in terms of binomial coeffic

Discrete Chebyshev polynomials

In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev and rediscovered by Gra

Sieved Pollaczek polynomials

In mathematics, sieved Pollaczek polynomials are a family of sieved orthogonal polynomials, introduced by . Their recurrence relations are a modified (or "sieved") version of the recurrence relations

Turán's inequalities

In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Pál Turán (and first published by ). There are many generalizations to other polynomials, often called Turá

Heckman–Opdam polynomials

In mathematics, Heckman–Opdam polynomials (sometimes called Jacobi polynomials) Pλ(k) are orthogonal polynomials in several variables associated to root systems. They were introduced by Heckman and Op

Little q-Jacobi polynomials

In mathematics, the little q-Jacobi polynomials pn(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . Roelof Koekoek, Peter A. Lesky, and R

Q-Krawtchouk polynomials

In mathematics, the q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw . give a detail

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 recurr

Geronimus polynomials

In mathematics, Geronimus polynomials may refer to one of the several different families of orthogonal polynomials studied by Yakov Lazarevich Geronimus.

Jacobi polynomials

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight on the interval . Th

Q-Hahn polynomials

In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give a detailed list

Stieltjes–Wigert polynomials

In mathematics, Stieltjes–Wigert polynomials (named after Thomas Jan Stieltjes and Carl Severin Wigert) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, for the w

Al-Salam–Ismail polynomials

In mathematics, the Al-Salam–Ismail polynomials are a family of orthogonal polynomials introduced by Al-Salam and Ismail.

Continuous Hahn polynomials

In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeome

Orthogonal polynomials

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. The most wid

Bateman polynomials

In mathematics, the Bateman polynomials are a family Fn of orthogonal polynomials introduced by Bateman. The Bateman–Pasternack polynomials are a generalization introduced by . Bateman polynomials can

Christoffel–Darboux formula

In mathematics, the Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin Bruno Christoffel and Jean Gaston Darboux. It states that where fj(x) is th

Associated Legendre polynomials

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation or equivalently where the indices ℓ and m (which are integers) are referred to as the d

Continuous q-Hahn polynomials

In mathematics, the continuous q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw give a de

Meixner–Pollaczek polynomials

In mathematics, the Meixner–Pollaczek polynomials are a family of orthogonal polynomials P(λ)n(x,φ) introduced by Meixner, which up to elementary changes of variables are the same as the Pollaczek pol

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