The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function (Q1392788)

Explicit formulae for the ultraspherical coefficients of the moments \(x^l f^{(q)}(x)\) of a general order derivative of the infinitely differentiable function \(f\) in terms of its ultraspherical coefficients are derived. In particular, formulae for the ultraspherical coefficients for the moments of one single ultraspherical polynomial of certain degree are given. The author has shown how these relations can be applied to solving ordinary differential equations with polynomial coefficients.