Explicit Formulae for the Coefficients of Integrated Expansions of Jacobi Polynomials and Their Integrals

Abstract

A new formula expressing explicitly the integrals of Jacobi polynomials of any degree and for any order in terms of the Jacobi polynomials themselves is proved. Another new explicit formula relating the Jacobi coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established. The results for ultraspherical polynomials, of which the Chebyshev and Legendre polynomials are important special cases, are noted. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.

Keywords:

• Jacobi Polynomials
• Spectral Methods
• Expansion Coefficients
• Singular Sturm-Liouville Problem