The C-Ring -- Certain polynomial functions on -- An application of the campbell-hausdorff formula -- The โFundamental theorem of calculusโ 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

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