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

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 \[ 1+\frac{qz|}{|1}+\frac{q^{2}z|}{|1}+\frac{q^{3}z|}{|1}+\cdots \] provides a counterexample to the conjecture. We also highlight some other interesting phenomena displayed by this fraction.

Authors

Doron S. Lubinsky

The John Knopfmacher Centre
University of Witwatersrand
Wits, South Africa
and
School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332
United States