Rogers-Ramanujan and the Baker-Gammel-Wills (Pade) conjecture

• 成果类型: Article
• 署名作者: Lubinsky, DS
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2003.157.847
• 发表日期: 2003
• 页码: 847-889
• 关键词: Polynomials; approximants; CONVERGENCE; series

摘要

1961, Baker, Gammel and Wills conjectured that for functions f meromorphic in the unit ball, a subsequence of its diagonal Pade approximants converges uniformly in compact subsets of the ball omitting poles of f. There is also apparently a cruder version of the conjecture due to Pade himself, going back to the early twentieth century. We show here that for carefully chosen q on the unit circle, the Rogers-Ramanujan continued fraction 1+ (qz)\/(\1) + q(2z\)/\1 + q(3z)\/(\1) + ... provides a counterexample to the conjecture. We also highlight some other interesting phenomena displayed by this fraction.