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Integrability and geometry of the Wynn recurrence

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Abstract

We show that the Wynn recurrence (the missing identity of Frobenius of the Padé 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 Padé 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.

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Notes

  1. We thank Jarosław Kosiorek for pointing us such a geometric interpretation of the construction.

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Acknowledgements

The authors would like to thank the reviewers for their constructive comments, which allowed for the improvement of the presentation of the results.

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  1. Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. Słoneczna 54, 10-710, Olsztyn, Poland

    Adam Doliwa & Artur Siemaszko

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  1. Adam Doliwa
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Doliwa, A., Siemaszko, A. Integrability and geometry of the Wynn recurrence. Numer Algor 92, 571–596 (2023). https://doi.org/10.1007/s11075-022-01344-5

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