Solutions of the equation $a_n + (a_{n-1} + \cdots (a_2 + (a_1 + x^{r_1})^{r_2}\cdots )^{r_{n}} = b\, x$

TL;DR

This paper introduces a new upper bound for real solutions of a complex nested equation using a generalized algorithm, and develops Chebyshev functions tailored for this problem.

Contribution

It presents a novel upper bound and a specialized set of Chebyshev functions for solving a complex nested algebraic equation.

Findings

Established a new upper bound for real solutions.

Developed a set of Chebyshev functions adapted to the problem.

Demonstrated the effectiveness of the generalized derivation-division algorithm.

Abstract

We establish a novel upper bound for the real solutions of the equation specified in the title, employing a generalized derivation-division algorithm. As a consequence, we also derive a new set of Chebyshev functions adapted specifically for this problem.