The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function

A formula for the ultraspherical coefficients of the moments of one single ultraspherical polynomial of certain degree is given. Formulae for the ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its ultraspherical coefficients are also obtained. The corresponding formulae for the important special cases of Chebyshev polynomials of the first and second kinds and of Legendre polynomials are deduced. Two interesting numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations of lowest order in the ultraspherical expansion coefficients, in the sense of Lewanowicz, are discussed. Comparisons with the results obtained by optimal algorithm of Lewanowicz (1976) are also made.