Hirota’s bilinear method, Shanks’ transformation, and the ε-algorithms

Hirota's bilinear method can be quite useful in the solution of nonlinear differential and difference equations. In this paper, we show how this method can lead to a novel proof that the epsilon-algorithm of Wynn implements the Shanks' sequence transformation and, reciprocally, that the quantities it computes are expressed as ratios of Hankel determinants as given by Shanks. New identities between Hankel determinants and the quantities involved in Hirota's method are obtained, and they form the basis of our proof. Then, the same bunch of results is showed to hold also for the confluent form of the epsilon-algorithm. This treatment could also be useful for other sequence transformations and the corresponding recursive algorithms.