Carleman Linearization and Systems of Arbitrary Depth Polynomial Recursions

TL;DR

This paper introduces a novel approach using Carleman linearization to solve polynomial recurrence systems of arbitrary depth, providing explicit solutions and conditions for applicability.

Contribution

The paper develops a new method based on Carleman linearization for solving polynomial recursions of any depth, including reduction techniques and analysis of limitations.

Findings

Explicit solutions for polynomial recurrences derived

Conditions for the method's applicability established

Reduction of arbitrary depth systems to depth-one systems demonstrated

Abstract

New approach to systems of polynomial recursions is developed based on the Carleman linearization procedure. The article is divided into two main sections: firstly, we focus on the case of uni-variable depth-one polynomial recurrences. Subsequently, the systems of depth-one polynomial recurrence relations are discussed. The corresponding transition matrix is constructed and upper triangularized. Furthermore, the powers of the transition matrix are calculated using the back substitution procedure. The explicit expression for a solution to a broad family of recurrence relations is obtained. We investigate to which recurrences the framework can be applied and construct a sufficient conditions for the method to work. It is shown how introduction of auxiliary variables can be used to reduce arbitrary depth systems to the depth-one system of recurrences dealt with earlier. Finally, the limitations of the method are discussed, outlining possible directions of future research.