New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

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