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where S ⊂ Z is finite, cy : Z → C are a set of coefficient functions and g : Z → C. A solution of (1) is a function f : Z → C that satisfies the equation. Let l > 0. A lattice path with length l is a finite sequence p(0), p(1), . . . , p(L) of points in Z , and its steps are the set of lattice vectors {0} ∪ {p(k)− p(k − 1) : k = 1, . . . , l}. Specific classes of lattice paths arise by imposing some conditions on the paths: the steps are in a specified S ⊂ Z , the points are in a specified P ⊂ Z , the length L is fixed, and the points are distinct (non intersecting paths). In the context of lattice path counting problems the function f : Z → Z> that counts the number f(x) of paths in a specified class for which p(0) = 0 and p(L) = x is computed (the condition p(0) = 0 does not result in a loss of generality). In the context of restricted lattice class problems function f is computed for a class of paths whose points belong to a specified subset P ⊂ Z . Clearly 0 ∈ P otherwise there are no paths in P that start at 0. If the possible set of steps S ⊂ Z then counting function f has support in P, f(0) = 1, and f satisfies the linear homogeneous difference equation

Difference Equations and Generating Functions for some Lattice Path Problems