In this paper we study the Euclidean conformal covariant operator product expansion of one spinor field ψ(x) with dimensiond and of one vector field with dimension 1 in the gradient model in 4 dimensions and in the Thirring model in 2 dimensions. The fields contributing to the operator product can be determined by making the partial-wave analysis of the Schwinger functions of the theory in terms of operators belonging to irreducible representations of the conformal group and finding with the help of Ward identities the poles of the expansions in the dimensionl of these operators. If we consider operators with spin 1/2, there are in both models two poles: one forl=d, which is connected with the contribution of the fundamental field ψ(x), and one forl=d+1, which has a kinematical nature and can be represented in the operator product expansion by a field\(\psi _1 \left( x \right)\) with dimensiond+1. This, however, gives rise to problems of compatibility with the equation of motion of the field ψ(x). From this fact some interesting results about the models can be derived: closed equations for the Schwinger functions of the models and the relation between the dimensionsd of ψ(x) and the coupling constant.