Abstract
Formulae for the Legendre coefficients of the moments of the general order derivative of an infinitely differentiable function in terms of its Legendre coefficients are derived. Two 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 Legendre expansion coefficients are discussed. Comparisons with the results obtained by the optimal algorithm of Lewanowicz (1976) are made.