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

Details Rogers-Ramanujan and the Baker-Gammel-Wills (Padé) conjecture Rogers-Ramanujan and the Baker-Gammel-Wills (Padé) conjecture

2004-02-18

arxiv.org/abs/math/0402305v1

Ann. of Math. (2), Vol. 157 (2003), no. 3, 847--889

Mathematics

Classical Analysis and ODEs

43 pages published version

Scientific paper

In 1961, Baker, Gammel and Wills conjectured that for functions $f$ meromorphic in the unit ball, a subsequence of its diagonal Pad\'{e} 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 Pad\'{e} himself, going back to the earlier twentieth century. We show here that for carefully chosen $q$ on the unit circle, the Rogers-Ramanujan continued fraction $$1+\frac{qz|}{|1}+\frac{q^{2}z|}{|1}+\frac{q^{3}z|}{|1}+... $$ provides a counterexample to the conjecture. We also highlight some other interesting phenomena displayed by this fraction.

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