Discrete Boundary-value Problems
نویسندگان
چکیده
We consider multivariate sequences, deened by a partial linear recurrence equation together with appropriate boundary conditions. The domain of deenition of these sequences is the rst orthant of the integer lattice, restricted in some dimensions to an initial segment of the nonnegative integers. By means of the kernel method we obtain an explicit expression for the generating function of the solution in the case of a single restricted dimension, provided that the apex of the recurrence vanishes in all the remaining dimensions. Unlike the initial-value problem (where the domain is unrestricted in all dimensions) with apex of this type, this generating function depends rationally on the generating functions of the boundary conditions and of the right-hand side of the equation. As an example, we count lattice paths between two parallel hyperplanes with rational incline. By using the kernel method twice, we also solve a discrete Dirichlet problem in two dimensions.
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