Novel Formulas of Specific Non-Symmetric Jacobi Polynomials with an Application in Numerical Analysis

This paper introduces new formulas for non-symmetric Jacobi polynomials of specific parameters, focusing specifically on the subclasses where the difference between the two parameters of Jacobi polynomials is two or three. First, several key expressions of these polynomials are established, such as the power form expression and its inverse expression. After that, further essential formulas such as the derivatives of moments, linearization and connection formulas, and a formula for the repeated integrals are developed. Symbolic algebra is pivotal for summing some sums in closed forms. An application of some of the introduced formulas is included. The FitzHugh–Nagumo equation—a nonlinear differential equation arising in neuroscience—is solved using the collocation method. The presented numerical examples demonstrate the accuracy and efficiency of the proposed algorithm.