Convergence of the Rogers--Ramanujan continued fraction

V. I. Buslaev

Sbornik. Mathematics, Tome 194 (2003) no. 6, pp. 833-856

Résumé

Set $q=\exp(2\pi i\tau)$, where $\tau$ is an irrational number, and let $R_q$ be the radius of holomorphy of the Rogers–Ramanujan function $$ G_q(z)=1+\sum_{n=1}^\infty z^n\frac{q^{n^2}}{(1-q)\dotsb(1-q^n)}\,. $$ As is known, $R_q\leqslant 1$ and for each $\alpha\in[0,1]$ there exists $q=q(\alpha)$ such that $R_{q(\alpha)}=\alpha$. It is proved here that the function $H_q(z)=G_q(z)/G_q(qz)$ is meromorphic not only in the disc $=\{|z|$, but also in the disc $D=\{|z|1\}$, which is larger for $R_q1$; and that the Rogers–Ramanujan continued fraction converges to $H_q$ on compact subsets contained in $D\setminus\Omega_q$, where $\Omega_q$ is the union of circles with centres at $z=0$ and passing through the poles of $H_q$. The convergence of the Rogers–Ramanujan continued fraction in the domain $\Bigl\{|z|\max\bigl(R_q,\frac1{2+|1+q|}\bigr)\Bigr\}\setminus\Omega_q$ was established earlier by Lubinsky.