Rogers-Ramanujan and the Baker-Gammel-Wills (Padé) conjecture | Annals of Mathematics

Abstract In 1961, Baker, Gammel and Wills conjectured that for functions f meromorphic in the unit ball, a subsequence of its diagonal Padé 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 Padé 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+q2z||1+q3z||1+⋯ provides a counterexample to the conjecture. We also highlight some other interesting phenomena displayed by this fraction.

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数学年刊（Annals of Mathematics）

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