On the Rogers--Ramanujan Periodic Continued Fraction

V. I. Buslaev ; S. F. Buslaeva

Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 827-837

Résumé

In the paper, the convergence properties of the Rogers–Ramanujan continued fraction $$ 1+\frac{qz}{1+\frac{q^2z}{1+\cdots}} $$ are studied for $q=\exp (2\pi i\tau)$, where $\tau$ is a rational number. It is shown that the function $H_q$ to which the fraction converges is a counterexample to the Stahl conjecture (the hyperelliptic version of the well-known Baker–Gammel–Wills conjecture). It is also shown that, for any rational $\tau$, the number of spurious poles of the diagonal Padé approximants of the hyperelliptic function $H_q$ does not exceed one half of its genus.