Higher dimensional restricted lattice paths with diagonal steps

In this paper, restricted minimal lattice paths with horizontal, vertical, and diagonal steps, in two and higher dimensions are discussed. The Delannoy numbers, the numbers of unrestricted minimal lattice paths with diagonal steps, and some of their properties are introduced. The recurrence on the Delannoy numbers is extended to higher dimensions. The relation in two dimensions between the restricted minimal lattice paths and the Delannoy numbers is shown through the use of Andre's reflection principle. This relation is generalized from two dimensions to higher dimensions and is found to be in the form of a determinant. The relation between unrestricted and restricted weighted minimal lattice paths in two dimensions is shown by the extension of Andre's reflection principle.