Integrability and geometry of the Wynn recurrence

DOI10.1007/S11075-022-01344-5arXiv2201.01749WikidataQ114224267 ScholiaQ114224267MaRDI QIDQ6387589

Publication date: 5 January 2022

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.

Mathematics Subject Classification ID

Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis (42C05) Padé approximation (41A21) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions (37K20) Dynamical systems in numerical analysis (37N30) Lattice dynamics; integrable lattice equations (37K60) Configuration theorems in linear incidence geometry (51A20) Numerical analysis (65-XX) Numerical aspects of recurrence relations (65Q30)

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