Rogers-Ramanujan and the Baker-Gammel-Wills (Pad\'e) conjecture

TL;DR

This paper presents a counterexample to a longstanding conjecture about the convergence of Padé approximants, using the Rogers-Ramanujan continued fraction for specific values of q on the unit circle.

Contribution

The paper demonstrates that the Rogers-Ramanujan continued fraction provides a counterexample to the Baker-Gammel-Wills conjecture about Padé approximant convergence.

Findings

Counterexample to the conjecture using Rogers-Ramanujan continued fraction

Identification of phenomena exhibited by the Rogers-Ramanujan fraction

Insights into the behavior of Padé approximants for specific functions

Abstract

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.