Recursion formulas for integrated products of Jacobi polynomials

TL;DR

This paper derives recursion formulas for integrals of Jacobi polynomial products using hypergeometric series relations, enabling faster computations in numerical methods like FEM for PDEs.

Contribution

It introduces new contiguous relations for hypergeometric series that lead to efficient recursive computation of Jacobi polynomial integrals.

Findings

Recursive formulas significantly speed up integral calculations.

Application to FEM improves computational efficiency.

Numerical example confirms faster solution of PDE boundary value problems.

Abstract

From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric series. With these contiguous relations one can prove several recursion formulas of those series. This theoretical result allows to compute integrals over products of Jacobi polynomials in a very efficient recursive way. Moreover, the authors present an application to numerical analysis where it can be used in algorithms which compute the approximate solution of boundary value problem of partial differential equations by means of the finite elements method (FEM). With the aid of the contiguous relations, the approximate solution can be computed much faster than using numerical integration. A numerical example illustrates this effect.