Integrability and geometry of the Wynn recurrence

TL;DR

This paper links the Wynn recurrence to integrable systems and geometry, extending it to non-commutative settings and demonstrating applications in language theory and discrete analytic functions.

Contribution

It introduces a geometric and integrable systems perspective on the Wynn recurrence, including non-commutative generalizations and applications.

Findings

Wynn recurrence is a reduction of the discrete Schwarzian KP equation.

The geometric interpretation involves constrained quadrangular point sets.

Application to non-commutative Padé theory and discrete analytic functions.

Abstract

We show that the Wynn recurrence (the missing identity of Frobenius of the Pad\'{e} approximation theory) can be incorporated into the theory of integrable systems as a reduction of the discrete Schwarzian Kadomtsev-Petviashvili equation. This allows, in particular, to present the geometric meaning of the recurrence as a construction of the appropriately constrained quadrangular set of points. The interpretation is valid for a projective line over arbitrary skew field what motivates to consider non-commutative Pad\'{e} theory. We transfer the corresponding elements, including the Frobenius identities, to the non-commutative level using the quasideterminants. Using an example of the characteristic series of the Fibonacci language we present an application of the theory to the regular languages. We introduce the non-commutative version of the discrete-time Toda lattice equations together with their integrability structure. Finally, we discuss application of the Wynn recurrence in a different context of the geometric theory of discrete analytic functions.