汇报标题 (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卵大局以及离散可积系统：I,II,III)
汇报人 (Speaker)： Andrew N,W. Hone 教授（肯特大学，英国）
汇报功夫 (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
汇报地址 (Place)：校本部GJ303
约请人(Inviter)：张雄师 教授
主办部门：理学院数学系
汇报提要：
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.
