Jacobi polynomials

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) $P_{n}^{(\alpha ,\beta )}(x)$ are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight $(1-x)^{\alpha }(1+x)^{\beta }$ 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.

^ Szegő, Gábor (1939). "IV. Jacobi polynomials.". Orthogonal Polynomials. Colloquium Publications. Vol. XXIII. American Mathematical Society. ISBN 978-0-8218-1023-1. MR 0372517. 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; the asymptotic behavior is in VIII.2

[sz-1] Szegő, Gábor (1939). "IV. Jacobi polynomials.". Orthogonal Polynomials. Colloquium Publications. Vol. XXIII. American Mathematical Society. ISBN 978-0-8218-1023-1. MR 0372517. 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; the asymptotic behavior is in VIII.2

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