A Review on the Properties of Jacobi Polynomials.
Authors: Ben Hamdin Haniyah A. 
Published: February 2025

This review is dedicated to Jacobi polynomials which are a generalization of some well-known classical polynomials such as Gegenbauer polynomials, Legendre polynomials, first and second kinds Chebyshev polynomials and Zernike polynomials. To save effort and time, it is advantageous and sufficient to investigate the properties of Jacobi polynomials rather than considering their special cases separately. For demonstration purposes, we show how to reduce most of the obtained properties of Jacobi polynomials to the corresponding properties of Legendre polynomials such as the generating function, Rodrigues formula, special values and the orthogonality property. Most of the properties of Jacobi polynomials are obtained through their hypergeometric representations such as differential recurrence relations, generating functions, some special values (exact and asymptotic) and some integral expansions of positive integrand. The standard orthogonality property of Jacobi polynomials for the values of the parameters  is discussed and some applications of such important property are pointed out. The orthogonality property of Jacobi polynomials considerably affects the positions of their zeros. These zeros are essential in any type of numerical quadrature such as Legendre-Gaussian quadrature, first kind Chebyshev-Gaussian quadrature and second kind Chebyshev-Gaussian quadrature. Moreover the recurrence relations of Jacobi polynomials play an important role in computing the zeros of Jacobi polynomials which are needed in the Gaussian-Jacobi quadrature. A narrative review is provided on some attempts were done in extending the orthogonality property of Jacobi polynomials to non-standard values of the indexes  by amending some of the orthogonality conditions such as Sobolev and non-hermitian orthogonality. Integral expansions of Jacobi polynomials with a prominent feature (positive integrand) were presented in terms of other Jacobi polynomials. Such integral expansions should allow a variety of applications for Jacobi polynomials.

This review is dedicated to Jacobi polynomials which are a generalization of some well-known classical polynomials such as Gegenbauer polynomials, Legendre polynomials, first and second kinds Chebyshev polynomials and Zernike polynomials. To save effort and time, it is advantageous and sufficient to investigate the properties of Jacobi polynomials rather than considering their special cases separately. For demonstration purposes, we show how to reduce most of the obtained properties of Jacobi polynomials to the corresponding properties of Legendre polynomials such as the generating function, Rodrigues formula, special values and the orthogonality property. Most of the properties of Jacobi polynomials are obtained through their hypergeometric representations such as differential recurrence relations, generating functions, some special values (exact and asymptotic) and some integral expansions of positive integrand. The standard orthogonality property of Jacobi polynomials for the values of the parameters  is discussed and some applications of such important property are pointed out. The orthogonality property of Jacobi polynomials considerably affects the positions of their zeros. These zeros are essential in any type of numerical quadrature such as Legendre-Gaussian quadrature, first kind Chebyshev-Gaussian quadrature and second kind Chebyshev-Gaussian quadrature. Moreover the recurrence relations of Jacobi polynomials play an important role in computing the zeros of Jacobi polynomials which are needed in the Gaussian-Jacobi quadrature. A narrative review is provided on some attempts were done in extending the orthogonality property of Jacobi polynomials to non-standard values of the indexes  by amending some of the orthogonality conditions such as Sobolev and non-hermitian orthogonality. Integral expansions of Jacobi polynomials with a prominent feature (positive integrand) were presented in terms of other Jacobi polynomials. Such integral expansions should allow a variety of applications for Jacobi polynomials.