Jacobi polynomials

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