Polyharmonic Functions in the Quarter Plane

Abstract

While discrete harmonic functions have been objects of interest for quite some time, this is not the case for discrete polyharmonic functions, as appear for instance in the asymptotics of path counting problems. In this article, a novel method to compute all discrete polyharmonic functions in the quarter plane for non-singular models with small steps and zero drift is proposed. In case of a finite group, an alternative method using decoupling functions is given, which often leads to a basis consisting of rational functions. In a similar manner one can obtain polyharmonic functions in the continuous setting, and convergence between the discrete and continuous cases is proven. Lastly, using a concrete example it is shown why the decoupling approach seems not to work in the infinite group case.