A Family of Generalized Jacobi Polynomials

A Family of Generalized Jacobi Polynomials

The family of orthogonal polynomials corresponding to a generalized Jacobi weight function was considered by Wheeler and Gautschi who derived recurrence relations, both for the related Chebyshev moments and for the associated orthogonal polynomials. We obtain an explicit representation of these polynomials, from which the recurrence relation can be derived.

Generalized Jacobi polynomials/functions and their applications

We introduce a family of generalized Jacobi polynomials/functions with indexes α,β ∈ R which are mutually orthogonal with respect to the corresponding Jacobi weights and which inherit selected important properties of the classical Jacobi polynomials. We establish their basic approximation properties in suitably weighted Sobolev spaces. As an example of their applications, we show that the gener...

An Integral Formula for Generalized Gegenbauer Polynomials and Jacobi Polynomials

Algebraic properties of a family of Jacobi polynomials

The one-parameter family of polynomials Jn(x, y) = ∑n j=0 ( y+j j ) x is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each n ≥ 6, the polynomial Jn(x, y0) is irreducible over Q for all but finitely many y0 ∈ Q. If n is odd, then with the exception of a finite set of y0, the Galois group of Jn(x, y0) is Sn; if n is even, then the exceptional set is thin.

Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials

We extend the definition of the classical Jacobi polynomials withindexes α,β > −1 to allow α and/or β to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the...