  The purpose of this paper is to show and explain a new formula that indicates
with finality the derivatives of Shifted Monic Ultraspherical polynomials
(SMUPs) of any degree and for any fractional-order using the shifted Monic
Ultraspherical polynomials themselves. We also create a direct method solution
for the linear or nonlinear multi-order fractional differential equations
(FDEs) with constant coefficients involving a spectral Galerkin method. The
spatial approximation with its fractional order derivatives (described in the
Caputo sense) are built using shifted Monic Ultraspherical polynomials.
