Wynn递推的可积性与几何
Integrability and geometry of the Wynn recurrence
AI总结：
本文将Wynn递推视为离散Schwarzian KP方程的约化，揭示其射影几何意义，并建立非交换Padé理论、非交换Toda方程及其在正则语言和离散解析函数中的应用。
摘要：
我们证明了Wynn递推（Padé逼近理论中缺失的Frobenius恒等式）可以作为离散Schwarzian Kadomtsev-Petviashvili方程的一个约化而纳入可积系统理论。这尤其使得可以把该递推的几何意义表述为构造一个受到适当约束的四点集。该解释对任意斜域上的射影直线均成立，从而激发了对非交换Padé理论的研究。我们利用拟行列式把包括Frobenius恒等式在内的相应元素转移到非交换层面。通过Fibonacci语言的特征级数这一例子，我们展示了该理论在正则语言中的应用。我们引入了离散时间Toda格方程的非交换版本及其可积性结构。最后，我们讨论了Wynn递推在离散解析函数几何理论这一不同背景下的应用。
英文摘要：
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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