# Category: Orthogonal polynomials

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
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.
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
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