Skew-Orthogonal Polynomials and Pfaff Lattice Hierarchy Associated With an Elliptic Curve (2024)

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Volume 2024 Issue 10 May 2024
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Wei Fu

School of Mathematical Sciences, Key Laboratory of Mathematics and Engineering Applications (Ministry of Education) & Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University

, Shanghai 200241, People’s Republic of

China

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Shi-Hao Li

School of Mathematics, Sichuan University

, Chengdu 610064, People’s Republic of

China

Correspondence to be sent to: e-mail: lishihao@lsec.cc.ac.cn

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International Mathematics Research Notices, Volume 2024, Issue 10, May 2024, Pages 8695–8715, https://doi.org/10.1093/imrn/rnad305

Published:

24 January 2024

Article history

Received:

21 July 2023

Revision received:

21 July 2023

Published:

24 January 2024

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    Wei Fu, Shi-Hao Li, Skew-Orthogonal Polynomials and Pfaff Lattice Hierarchy Associated With an Elliptic Curve, International Mathematics Research Notices, Volume 2024, Issue 10, May 2024, Pages 8695–8715, https://doi.org/10.1093/imrn/rnad305

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Abstract

Starting with a skew-symmetric inner product over an elliptic curve, we propose the concept of elliptic skew-orthogonal polynomials. Inspired by the Landau–Lifsh*tz hierarchy and its corresponding time evolutions, we obtain the recurrence relation and the |$\tau $|-function representation for such a novel class of skew-orthogonal polynomials. Furthermore, a bilinear integral identity is derived through the so-called Cauchy–Stieljes transformation, from which we successfully establish the connection between the elliptic skew-orthogonal polynomials and an elliptic extension of the Pfaff lattice hierarchy.

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