Multistep epsilon-algorithm, Shanks' transformation, and Lotka-Volterra system by Hirota's method

TL;DR

This paper introduces a multistep extension of Wynn's epsilon-algorithm and Shanks' transformation, linking them to an extended Lotka-Volterra system using Hirota's bilinear method, advancing nonlinear difference equation analysis.

Contribution

It develops a multistep epsilon-algorithm and demonstrates its equivalence to a multistep Shanks' transformation, connecting these to an extended Lotka-Volterra system via Hirota's method.

Findings

Multistep epsilon-algorithm implemented for sequence transformation.

Connection established between the algorithm and an extended Lotka-Volterra system.

Hirota's bilinear method used to derive and analyze the system.

Abstract

In this paper, we give a multistep extension of the epsilon-algorithm of Wynn, and we show that it implements a multistep extension of the Shanks' sequence transformation which is defined by ratios of determinants. Reciprocally, the quantities defined in this transformation can be recursively computed by the multistep epsilon-algorithm. The multistep epsilon-algorithm and the multistep Shanks' transformation are related to an extended discrete Lotka-Volterra system. These results are obtained by using the Hirota's bilinear method, a procedure quite useful in the solution of nonlinear partial differential and difference equations.