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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 \[ 1+\frac{qz|}{|1}+\frac{q^{2}z|}{|1}+\frac{q^{3}z|}{|1}+\cdots \] provides a counterexample to the conjecture. We also highlight some other interesting phenomena displayed by this fraction.
@article{10_4007_annals_2003_157_847,
author = {Doron S. Lubinsky},
title = {Rogers-Ramanujan and the {Baker-Gammel-Wills} {(Pad\'e)} conjecture},
journal = {Annals of mathematics},
pages = {847--889},
publisher = {mathdoc},
volume = {157},
number = {3},
year = {2003},
doi = {10.4007/annals.2003.157.847},
mrnumber = {1983783},
zbl = {1068.41033},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.157.847/}
}
TY - JOUR AU - Doron S. Lubinsky TI - Rogers-Ramanujan and the Baker-Gammel-Wills (Padé) conjecture JO - Annals of mathematics PY - 2003 SP - 847 EP - 889 VL - 157 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.157.847/ DO - 10.4007/annals.2003.157.847 LA - en ID - 10_4007_annals_2003_157_847 ER -
%0 Journal Article %A Doron S. Lubinsky %T Rogers-Ramanujan and the Baker-Gammel-Wills (Padé) conjecture %J Annals of mathematics %D 2003 %P 847-889 %V 157 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.157.847/ %R 10.4007/annals.2003.157.847 %G en %F 10_4007_annals_2003_157_847
Doron S. Lubinsky. Rogers-Ramanujan and the Baker-Gammel-Wills (Padé) conjecture. Annals of mathematics, Tome 157 (2003) no. 3, pp. 847-889. doi: 10.4007/annals.2003.157.847
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