Polynomial Fermionic Forms for the Branching Functions of the Rational Coset Conformal Field Theories $\widehat{su}(2)_{M}\times \widehat{su}(2)_{N}/\widehat{su}(2)_{M+N}$}

General fermionic expressions for the branching functions of the rational coset conformal field theories $\widehat{su}(2)_{M}\times \widehat{su}(2)_N/\widehat{su}(2)_{M+N}$ are given. The equality of the bosonic and fermionic representations for the branching functions is proven by introducing polynomial truncations of these branching functions which are the configuration sums of the RSOS models in regime III. The path space interpretation of the RSOS models provides recursion relations for the configuration sums. The proof of the recursion relations for the fermionic expressions is given by using telescopic expansion techniques. The configuration sums of the RSOS model in regime II which correspond to the branching functions of the $Z_{M+N}$-parafermion conformal field theory are obtained by the duality transformation $q\rightarrow q^{-1}$.