In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number...

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation ( 1 − x 2 ) d 2 d x 2 P ℓ m ( x ) − 2...

mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. He is also known for his...

with Gauss–Legendre quadrature, the associated orthogonal polynomials are Legendre polynomials, denoted by Pn(x). With the n-th polynomial normalized...

Qn, are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the...

(1 − x2)α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold...

left-hand side is played by the Legendre polynomials. The addition theorem states where Pℓ is the Legendre polynomial of degree ℓ. This expression is...

In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to...

mathematician Associated Legendre polynomials Legendre's equation Legendre polynomials Legendre symbol Legendre transformation Legendre (crater), a lunar impact...

well-approximated by polynomials on [ − 1 , 1 ] {\displaystyle [-1,1]} , the associated orthogonal polynomials are Legendre polynomials, denoted by Pn(x)...

(including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials). They have many important applications in such...

in either the passband or stopband. The filter design is based on Legendre polynomials which is the reason for its alternate name and the "L" in Optimum...

Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. The Jacobi polynomials were...

formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was independently introduced by Olinde Rodrigues (1816), Sir...

In mathematics, the big q-Legendre polynomials are an orthogonal family of polynomials defined in terms of Heine's basic hypergeometric series as P n (...

the Legendre polynomials. Another collection of orthogonal polynomials are the associated Legendre polynomials. The study of orthogonal polynomials involves...

mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials P n ( x ) {\displaystyle...

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)}...

Legendre polynomials Associated Legendre polynomials Spherical harmonic Lucas polynomials Macdonald polynomials Meixner polynomials Necklace polynomial Newton...

to define the multidimensional polynomials. Like the other classical orthogonal polynomials, the Hermite polynomials can be defined from several different...

coefficients Pn are the Legendre polynomials of degree n. Therefore, the Taylor coefficients of the integrand are given by the Legendre polynomials in X = cos θ...

proceed to calculate the elliptic integral. Given Eq. 3 and the Legendre polynomial solution for the elliptic integral: K ( k ) = π 2 ∑ n = 0 ∞ ( ( 2...

equal to zero are proportional to spherical harmonics and associated Legendre polynomials, normalized to unity and with Condon and Shortley phase convention:...

are obtained by composing the Cayley transform with Legendre polynomials. A rational Legendre function of degree n is defined as: R n ( x ) = 2 x + 1 P n...

generalized harmonic mean estimator, and importance sampling. The Legendre polynomials are characterized by orthogonality with respect to the uniform measure...

supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets. Legendre functions have widespread applications...

Laguerre polynomials Chebyshev polynomials Legendre polynomials Jacobi polynomials Others come from statistics: Hermite polynomials Many are studied in algebra...

of latitude and may be expressed as an infinite sum of associated Legendre polynomials; the functions are orthogonal over the sphere in the continuous case...

{R}}_{i}{\hat {R}}_{j}\ ,} which makes more explicit the connection to Legendre polynomials which result from the multipole expansion, namely here P 2 ( x )...

In mathematics, the continuous q-Legendre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme.Koekoek, Lesky...