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


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.


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