M. Matsumoto characterized the Cartan connection \(C\Gamma\) of a Finsler space (M,L) by the vanishing of \[ (*)\quad g_{ij| k},\quad g_{ij}|_ k,\quad D^ i_ k,\quad T_ j^ i{}_ k,\quad S_ j^ i{}_ k \] (h- and v-covariant derivatives of \(g_{ij}\), the deflection tensor, and two torsion tensors). In this paper the coefficients \((N^ i_ k\), \(F_ j^ i{}_ k\), \(C_ j^ i{}_ k)\) of an arbitrary Finsler connection \(F\Gamma\) are expressed explicitly by the values of (*) taken for this \(F\Gamma\). Conversely, any system of values of (*) belongs to a well-determined \(F\Gamma\). This shows the excellence of Matsumoto's axioms for the \(C\Gamma\). Similar results are obtained for an \(F\Gamma\) in the case when \(C\Gamma\) and Matsumoto's axioms are replaced by the Berwald connection \(B\Gamma\) and by Okada's axioms for this \(B\Gamma\). The investigations are also extended to R. Miron's generalized Finsler spaces (M,g) (in which \(g_{ij}\) is not necessarily deduced from a fundamental function L).