Jacobi Polynomial
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A Jacobi Polynomial is an orthogonal polynomial that ...
- AKA: Hypergeometric Polynomial.
- Example(s):
- $P_0^{(\alpha,\beta)}(x) = 1$
- $P_1^{(\alpha,\beta)}(x) = \dfrac{1}{2}[2(\alpha+1)+(\alpha+\beta+2)(x-1)]$
- $P_2^{(\alpha,\beta)}(x) = \dfrac{1}{8}[4(\alpha+1)(\alpha+2)+4(\alpha+\beta+3)(\alpha+2)(x-1)+(\alpha+\beta+3(\alpha+\beta+4)(x-1)^2]$
- Counter-Example(s):
- See: Orthogonal Polynomials, Carl Gustav Jacob Jacobi, Mathematics, Classical Orthogonal Polynomials, Gegenbauer Polynomials, Legendre Polynomials, Zernike Polynomials, Chebyshev Polynomials.
References
2021a
- (Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Jacobi_polynomials Retrieved:2021-9-12.
- In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1 − x)α(1 + x)β on the interval [−1, 1]. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.[1]
The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.
- In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1 − x)α(1 + x)β on the interval [−1, 1]. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.[1]
- ↑ The definition is in IV.1; the differential equation – in IV.2; Rodrigues' formula is in IV.3; the generating function is in IV.4; the recurrent relation is in IV.5.
2021b
- (MathWorld, 2021) ⇒ https://mathworld.wolfram.com/JacobiPolynomial.html
- QUOTE: The Jacobi polynomials, also known as hypergeometric polynomials, occur in the study of rotation groups and in the solution to the equations of motion of the symmetric top. They are solutions to the Jacobi differential equation, and give some other special named polynomials as special cases. They are implemented in the Wolfram Language as JacobiP[n, a, b, z].