A solvable nonlinear autonomous recursion of arbitrary order

TL;DR

This paper presents a method to explicitly solve a class of nonlinear autonomous recursions of arbitrary order using algebraic operations, including solving linear systems and polynomial equations, extending solvability to higher orders.

Contribution

It introduces a general solution approach for nonlinear autonomous recursions of any order, involving algebraic methods and polynomial equations, which was not previously established.

Findings

Recursion of any order p can be solved explicitly.

Solution involves linear algebra and polynomial equations.

Applicable to complex-valued recursions.

Abstract

The initial-values problem of the following nonlinear autonomous recursion of order p , z (s + p) = c product of [z (s + l)]^a_l ; with p an arbitrary positive integer, z (s) the dependent variable (possibly a complex number), s the independent variable (a non negative integer), c an arbitrarily assigned, possibly complex, number, and the p exponents a_l arbitrarily assigned integers (positive, negative or vanishing, so that the right-hand side of the recursion be univalent)|is solvable by algebraic operations, involving the solution of a system of linear algebraic equations (generally explicitly solvable) and of a single polynomial equation of degree p (hence explicitly solvable for p = 1; 2; 3; 4 ).