Some Results on the Coefficients of Integrated Expansions of Ultraspherical Polynomials and their Integrals

The formula of expressing the coefficients of an expansion of ultraspherical polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is stated in a more compact form and proved in a simpler way than the formula of Phillips and Karageorghis (1990). A new formula is proved for the q times integration of ultraspherical polynomials, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.