Closed-Form Solutions for 12th-Order Nonlinear Rational Recursive Difference Equations

Abstract

This study presents direct computations of numerous closed-form solutions for
various rational recursive problems. Several results related to the ensuing rational recursive
sequences are examined in this article:

_{\[
\eta_{n+1} = \frac{\eta_{n-11}}{\pm 1 \pm \eta_{n-1}\eta_{n-3}\eta_{n-5}\eta_{n-7}\eta_{n-9}\eta_{n-11}},
\qquad n = 0,1,2,\ldots
\]}

where the initial conditions are arbitrary real numbers.
A universal closed-form solution for nonlinear rational recursive difference equations is
unlikely because of their inherent specificity and complexity; only certain forms can be solved
analytically. Researchers therefore focus on particular subclasses of these equations, often
transforming them into more tractable linear forms through appropriate substitutions. By
analyzing the structure of a given equation and applying the necessary transformations, we
can identify a solvable class of nonlinear difference equations and obtain a closed-form
solution.