Solving Discrete Initial- and Boundary-Value Problems

Multivariate linear recurrences appear in such diverse elds of mathematics as combinatorics, probability theory, and numerical resolution of partial diierential equations. Whereas in the univariate case the solution of a constant-coeecient recurrence always has a rational generating function, this is no longer true in the multivariate case where this generating function can be non-rational, non-algebraic, and even non-D-nite. Nevertheless, there are important cases where the solution can be computed exactly in terms of algebraic functions. Examples include many lattice-path problems such as the enumerations of Dyck, Motzkin, and Schroeder paths, determining the cardinality of various free algebras, and (in some cases) the enumeration of permutations with a forbidden pattern.