Recursive initial value problems for Sheffer sequences

We consider certain systems of linear operator equations where the n-th order solution is made unique by an initial value taken from the preceeding solution. Such conditions arise in the enumeration of lattice paths with weighted left turns, if the paths are restricted by a lower linear barrier of integer slope. 1 Introduction All our lattice paths take unit steps in the " North (= y) and ! East (= x) direction, starting at the origin. Symmetry lies at the center of most of the appealing formulas in lattice paths enumeration, and some symmetries are still preserved when we enumerate paths with weighted left turns " ! . (x; y) " ! ! ! " (0; 0) ! Lattice Path with Weight 2 However, for such paths the two basic problems of staying below and above a line parallel to the diagonal are no longer symmetric (formulas (7) and (8)). The discrepancy between “below”and “above”gets considerably worse when the line has a slope di¤erent from one (formula (6) and Theorem 2). Yet the proof of Theorem 2 shows that there is still enough symmetry left in the recurrence relation to allow signi…cant simpli…cations. We enlist the help of the Finite Operator Calculus [3] to solve these problems. Suppose, B is a degree reducing linear operator on the algebra of polynomials,