The coefficients of differentiated expansions of double and triple Jacobi polynomials

Abstract. Formulae expressing explicitly the coecients of an ex-pansion of double Jacobi polynomials which has been partially dif-ferentiated an arbitrary number of times with respect to its vari-ables in terms of the coecients of the original expansion are stated and proved. Extension to expansion of triple Jacobi polynomials is given. The results for the special cases of double and triple ultra-spherical polynomials are considered. Also the results for Cheby-shev polynomials of the rst, second, third and fourth kinds and of Legendre polynomials are noted. An application of how to use double Jacobi polynomials for solving Poisson's equation in two variables subject to nonhomogeneous mixed boundary conditions is described. 1