Multistep epsilon-algorithm, Shanks' transformation, and Lotka-Volterra system by Hirota's method
Provides a theoretical unification of sequence transformations and integrable systems, but the results are primarily mathematical and incremental.
The paper extends Wynn's epsilon-algorithm to a multistep version that implements a multistep Shanks transformation, linking it to an extended discrete Lotka-Volterra system via Hirota's bilinear method.
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.