On the coefficients of differentiated expansions and derivatives of Jacobi polynomials

A formula expressing explicitly the derivatives of Jacobi polynomials of any degree and for any order in terms of the Jacobi polynomials themselves is proved. Another explicit formula, which expresses the Jacobi expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original Jacobi coefficients, is also given. The results for the special case of ultraspherical polynomials are considered. The results for Chebyshev polynomials of the first and second kinds and for Legendre polynomials are also noted. An application of how to use Jacobi polynomials for solving ordinary and partial differential equations is described.