This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

The primary focus of this article is on applying specific generalized Jacobi polynomials (GJPs) as basis functions to obtain the solution of linear and non-linear even-order two-point BVPs. These GJPs are orthogonal polynomials that are expressed as Legendre polynomial combinations. The linear even-order BVPs are treated using the Petrov–Galerkin method. In addition, a formula for the first-order derivative of these polynomials is expressed in terms of their original ones. This relation is the key to constructing an operational matrix of the GJPs that can be used to treat the non-linear two-point BVPs. In fact, a numerical approach is proposed using this operational matrix of derivatives to convert the non-linear differential equations into effectively solvable non-linear systems of equations. The convergence of the proposed generalized Jacobi expansion is investigated. To show the precision and viability of our suggested algorithms, some examples are given.