Rogers - Ramanujan and the Baker - Gammel - Wills ( Padé ) conjecture
نویسندگان
چکیده
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.
منابع مشابه
Reflections on the Baker-Gammel-Wills (Padé) Conjecture
In 1961, Baker, Gammel and Wills formulated their famous conjecture that if a function f is meromorphic in the unit ball, and analytic at 0, then a subsequence of its diagonal Padé approximants converges uniformly in compact subsets to f . This conjecture was disproved in 2001, but it generated a number of related unresolved conjectures. We review their status.
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