Rogers-Ramanujan and the Baker-Gammel-Wills (Pade") conjecture

1961, Baker, Gammel and Wills conjectured that for functions f meromorphic in the unit ball, a subsequence of its diagonal Pade 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 Pade 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) + q(2z\)/\1 + q(3z)\/(\1) + ... provides a counterexample to the conjecture. We also highlight some other interesting phenomena displayed by this fraction.