A wide class of partial functional equations (all generalizing in a quite natural way the classical recurrence relation for the $\Gamma$-function), is considered in § 1, in order to give an existence and unicity theorem for the solution of each of them, provided this solution be a partially lg-convex one. Further, in § 2 is sketched a straightforward method of resolution for an interesting subclass of such a class of equations, only involving simple techniques of finite differences calculus.