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

In this paper, we propose a multistep extension of the Shanks sequence transformation. It is defined as a ratio of determinants. Then, we show that this transformation can be recursively implemented by a multistep extension of the $\varepsilon$-algorithm of Wynn. Some of their properties are specified. Thereafter, the multistep $\varepsilon$-algorithm and the multistep Shanks transformation are proved to be related to an extended discrete Lotka-Volterra system. These results are obtained by using Hirota's bilinear method, a procedure quite useful in the solution of nonlinear partial differential and difference equations.