Some results on the coefficients of integrated expansions of ultraspherical polynomials and their integrals (Q2778246)

The authors obtain 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. It is stated in more compact form and proved in a simpler way than the formula of \textit{T.~N.~Phillips} and \textit{A.~Karageorghis} [SIAM J. Numer. Anal. 27, No. 3, 823-830 (1990; Zbl 0701.33007)]. NEWLINENEWLINENEWLINESome properties of Jacobi and ultraspherical polynomials are investigated. New formulae for integration of ultraspherical polynomials of arbitrary degree that have been integrated an arbitrary number of times in terms of ultraspherical polynomials themselves are also derived. The corresponding results for Chebyshev polynomials of first and second kinds and for Legendre polynomials are obtained. NEWLINENEWLINENEWLINEThe authors also explain how these formulae may be used for solving differential equations with varying coefficients. NEWLINENEWLINENEWLINEThe article contains an extended bibliography which reflects several results obtained in this area.