  Let (M_1,F_1) and (M_2,F_2) be a pair of Finsler manifolds. The Minkowskian
product Finsler manifold (M,F) of (M_1,F_1) and (M_2,F_2) with respect to a
product function f is the product manifold M=M_1\times M_2 endowed with the
Finsler metric F^2=f(K,H), where K=(F_1)^2,H=(F_2)^2. In this paper, the Cartan
connection and Berwald connection of (M,F) are derived in terms of the
corresponding objects of (M_1,F_1) and (M_2,F_2). Necessary and sufficient
conditions for (M,F) to be Berwald (resp. weakly Berwald, Landsberg, weakly
Landsberg) manifold are obtained. Thus an effective method for constructing
special Finsler manifolds mentioned above is given.
