Some Cauchy integral formulas for the null solutions of polynomial Dirac operators p(D) in the homogeneous space Rn are derived. As application, they are used to generalize Riesz-Dunford calculus. Precisely, they are used to the functional calculi for the n-tuple non-commuting bounded operators A under the conditions σ(< A, ξ>) Є R for allξЄ Rn, i.e. The operator-valued distributions for the kernel functions are constructed by both Weyl calculus and polynomial approaches. The monogenic functional calculus and the connections with Weyl calculus are to be discussed in details.