»ã±¨±êÌâ (Title)£ºElliptic & hyperelliptic analogues of Chebyshev polynomials, and related discrete integrable systems: I,II,III (ÇбÈÑ©·ò¶àÏîʽµÄÍÖÔ²ºÍ³¬ÂÑ´ó¾ÖÒÔ¼°ÀëÉ¢¿É»ýϵͳ£ºI,II,III)

»ã±¨ÈË (Speaker)£º Andrew N,W. Hone ½ÌÊÚ£¨¿ÏÌØ´óѧ£¬Ó¢¹ú£©

»ã±¨¹¦·ò (Time)£º

I: 2024Äê09ÔÂ25ÈÕ(ÖÜÈý) 14:00-15:30¡¢ 

II: 2024Äê09ÔÂ26ÈÕ(ÖÜËÄ) 14:00-15:30¡¢ 

III: 2024Äê09ÔÂ27ÈÕ(ÖÜÎå) 10:00-11:30

»ã±¨µØÖ· (Place)£ºÐ£±¾²¿GJ303

Ô¼ÇëÈË(Inviter)£ºÕÅÐÛʦ ½ÌÊÚ

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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.