We extend the definition of the classical Jacobi polynomials with
 indexes $\af,\bt>-1$ to allow $\af$ and/or $\beta$ to be negative
 integers.  We show that the generalized Jacobi polynomials, with
 indexes corresponding to the number of boundary conditions in a given
 partial differential equation, are the natural basis functions for
 the spectral approximation of this partial differential
 equation. Moreover, the use of generalized Jacobi polynomials leads
 to much simplified analysis, more precise error estimates and well
 conditioned algorithms.