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# Jacobi polynomials

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) are a class of classical orthogonal polynomials. [1]

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

## 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.

## Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

## Carl Gustav Jacob Jacobi

Carl Gustav Jacob Jacobi (10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.

## Chebyshev polynomials

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.

## 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).

## Continuous q-Jacobi polynomials

In mathematics, the continuous q-Jacobi polynomials P(x|q), introduced by, are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme.

## Domain (mathematical analysis)

In mathematical analysis, a domain is any connected open subset of a finite-dimensional vector space.

## Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as The value of each is taken to be 1 (an empty product) when n.

## Gamma function

In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.

## Gegenbauer polynomials

In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(x) are orthogonal polynomials on the interval with respect to the weight function (1 &minus; x2)α–1/2.

## Generating function

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a power series.

## Hypergeometric function

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases.

## Legendre polynomials

In mathematics, Legendre functions are solutions to Legendre's differential equation: They are named after Adrien-Marie Legendre.

## Linear differential equation

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where,..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable.

## 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.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## 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.

## Principal branch

In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function.

## 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 seems consistent to refer to them as Romanovski-Routh polynomials, by analogy with the terms Romanovski-Bessel and Romanovski-Jacobi used by Lesky.

## Rodrigues' formula

In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by, and.

## Romanovski polynomials

In mathematics, Romanovski polynomials is an informal term for one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski in French transcription) within the context of probability distribution functions in statistics.

## Zernike polynomials

In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk.

## References

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