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The C-Ring.- Certain polynomial functions on .- An application of the campbell-hausdorff formula.- The ¿Fundamental theorem of calculus¿ on .- Some estimates.- The representation q.- The differential equations satisfied by eiv-?(H(x)).- The differential equations on satisfied by the cusp form.- Application of the differential equations.- The functions FJ (? ) (b).- The homomorphism FJ.- Preservation of certain filtrations.- Injectivity of the homomorphism FJ.- Invariance properties of the homomorphism FJ(?).- Non-vanishing of certain polynomials.- Algebraic properties of the C-ring.- The difference equations satisfied by the C-function.- The critical points of the function ?(H( )).- The asymptotic behaviour of the C-function.- The uniqueness theorem.- The representation theorems.- Conclusion.