Constructing discrete harmonic functions in wedges

Abstract

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function. Contents 1. Introduction and main results 1 2. Study of the kernel 9 3. Boundary value problems for the generating functions 15 4. Proof of our main results (Theorems 1 and 2) 18 5. Various examples 25 References 35 Appendix A. Weierstrass' preparation and division theorems 38