- Algebraic structures
- >
- Lie groups
- >
- Hypergeometric functions
- >
- Special hypergeometric functions

- Algebraic structures
- >
- Lie groups
- >
- Special functions
- >
- Special hypergeometric functions

- Combinatorics
- >
- Special functions
- >
- Hypergeometric functions
- >
- Special hypergeometric functions

- Complex analysis
- >
- Special functions
- >
- Hypergeometric functions
- >
- Special hypergeometric functions

- Differential geometry
- >
- Lie groups
- >
- Hypergeometric functions
- >
- Special hypergeometric functions

- Differential geometry
- >
- Lie groups
- >
- Special functions
- >
- Special hypergeometric functions

- Discrete groups
- >
- Automorphic forms
- >
- Hypergeometric functions
- >
- Special hypergeometric functions

- Discrete mathematics
- >
- Combinatorics
- >
- Special functions
- >
- Special hypergeometric functions

- Fields of mathematical analysis
- >
- Complex analysis
- >
- Special functions
- >
- Special hypergeometric functions

- Fields of mathematics
- >
- Combinatorics
- >
- Special functions
- >
- Special hypergeometric functions

- Functions and mappings
- >
- Types of functions
- >
- Hypergeometric functions
- >
- Special hypergeometric functions

- Functions and mappings
- >
- Types of functions
- >
- Special functions
- >
- Special hypergeometric functions

- Harmonic analysis
- >
- Automorphic forms
- >
- Hypergeometric functions
- >
- Special hypergeometric functions

- Lie groups
- >
- Special functions
- >
- Hypergeometric functions
- >
- Special hypergeometric functions

- Manifolds
- >
- Lie groups
- >
- Hypergeometric functions
- >
- Special hypergeometric functions

- Manifolds
- >
- Lie groups
- >
- Special functions
- >
- Special hypergeometric functions

- Topological groups
- >
- Lie groups
- >
- Hypergeometric functions
- >
- Special hypergeometric functions

- Topological groups
- >
- Lie groups
- >
- Special functions
- >
- Special hypergeometric functions

- Types of functions
- >
- Special functions
- >
- Hypergeometric functions
- >
- Special hypergeometric functions

Neumann function

No description available.

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

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,

Confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three re

Lentz's algorithm

In mathematics, Lentz's algorithm is an algorithm to evaluate continued fractions and compute tables of spherical Bessel functions. The version usually employed now is due to Thompson and Barnett.

Elliptic integral

In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (c. 1750). T

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

Fresnel integral

The Fresnel integrals S(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (erf). They arise in the de

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

Bessel function

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation for an arbitrary complex

Exponential integral

In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument.

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

Sonine formula

In mathematics, Sonine's formula is any of several formulas involving Bessel functions found by Nikolay Yakovlevich Sonin. One such formula is the following integral formula involving a product of thr

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

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

Whittaker function

In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker to make the formulas involving the

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

Incomplete Bessel K function/generalized incomplete gamma function

Some mathematicians defined this type incomplete-version of Bessel function or this type generalized-version of incomplete gamma function:

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

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

Exponential function

The exponential function is a mathematical function denoted by or (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued functio

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

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

Wigner D-matrix

The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanic

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

Spherical harmonics

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific

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.

Error function

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as: This integral is a special (non-elementary) sig

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

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

Gamma function

In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function

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

Trigonometric integral

In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions.

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

Kelvin functions

In applied mathematics, the Kelvin functions berν(x) and beiν(x) are the real and imaginary parts, respectively, of where x is real, and Jν(z), is the νth order Bessel function of the first kind. Simi

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

Solid harmonics

In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions . There are two kinds: the regular solid harmonic

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

Airy function

In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(x) and

Logarithmic integral function

In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, accordi

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-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

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

Beta function

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by

Table of spherical harmonics

This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree . Some of these formulas are expressed in terms of the Cartesian expansion of the spherical ha

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

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

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

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

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

Coulomb wave function

In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb po

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

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,

Incomplete Bessel functions

In mathematics, the incomplete Bessel functions are types of special functions which act as a type of extension from the complete-type of Bessel functions.

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

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

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

Cunningham function

In statistics, the Cunningham function or Pearson–Cunningham function ωm,n(x) is a generalisation of a special function introduced by and studied in the form here by . It can be defined in terms of th

Zonal spherical harmonics

In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spheric

Bessel–Clifford function

In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alterna

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

Bateman function

In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman defined it by Bateman discovered this function,

Toronto function

In mathematics, the Toronto function T(m,n,r) is a modification of the confluent hypergeometric function defined by , Weisstein, as Later, Heatley (1964) recomputed to 12 decimals the table of the M(R

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

Parabolic cylinder function

In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation This equation is found when the technique of separation of variables is used on

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

© 2023 Useful Links.