报告题目 (Title)：Elliptic & hyperelliptic analogues of Chebyshev polynomials, and related discrete integrable systems: I,II,III (切比雪夫多项式的椭圆和超椭圆形式以及离散可积系统：I,II,III)

报告人 (Speaker)： Andrew N,W. Hone 教授（肯特大学，英国）

报告时间 (Time)：

I: 2024年09月25日(周三) 14:00-15:30、 

II: 2024年09月26日(周四) 14:00-15:30、 

III: 2024年09月27日(周五) 10:00-11:30

报告地点 (Place)：校本部GJ303

邀请人(Inviter)：张大军 教授

主办部门：理学院数学系

报告摘要：

Based on van der Poorten's work on continued fractions in function fields, we consider a family of orthogonal polynomials defined by the J-fraction expansion of a meromorphic function of order g+1 on a hyperelliptic curve of genus g. The case of a rational curve (g=0) just produces the Chebyshev polynomials of the 2nd kind, while the elliptic case (g=1) is related to elliptic orthogonal polynomials that were constructed by Akhiezer. For all g>0, the recurrence coefficients obey discrete dynamical systems which are algebraically integrable, being associated with genus g solutions of the Toda lattice. In particular, for g=1 we find a particular Quispel-Roberts-Thompson (QRT) map, together with explicit solutions in terms of Hankel determinants which satisfy the Somos-4 recurrence relation. If time permits, we will mention more recent results with Roberts, Vanhaecke and Zullo, relating to S-fraction expansions and solutions of Volterra/modified Volterra lattices.