Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan (Ramanujan J.) (20070101), 13, no.~1-3, 63-101. ISSN: 1382-4090 (print).eISSN: 1572-9303.
Subject
11 Number theory -- 11A Elementary number theory 11A55 Continued fractions
11 Number theory -- 11Y Computational number theory 11Y16 Algorithms; complexity
37 Dynamical systems and ergodic theory -- 37C Smooth dynamical systems: general theory 37C99 None of the above, but in this section
37 Dynamical systems and ergodic theory -- 37E Low-dimensional dynamical systems 37E99 None of the above, but in this section
Let $$ \scr{R}_1(a,b)\coloneq\frac{a}{1+\scr{S}(a,b)} \coloneq\cfrac a\\ 1+\cfrac b^2\\ 1+\cfrac4a^2\\ 1+\cfrac9b^3\\ 1+\ddots \endcfrac\quad . $$ The authors continue the investigation of $\scr{R}_{1}(a,b)$ begun in [J. M. Borwein, R. E. Crandall\ and G. J. Fee, Experiment. Math. {\bf 13} (2004), no.~3, 275--285; MR2103326 (2005g:11126)] and [J. M. Borwein\ and R. E. Crandall, Experiment. Math. {\bf 13} (2004), no.~3, 287--295; MR2103327 (2005h:11149)]. \par Define $$ \aligned \scr{D}_0&\coloneq \{(a,b)\in\Bbb{C}\times\Bbb{C}\colon\ \scr{R}_1(a,b)\ {\rm converges\ on}\ \widehat{\Bbb{C}}\},\\ \scr{D}_2&\coloneq \{(a,b)\in\Bbb{C}\times\Bbb{C}\colon\ |a|\not=|b|\},\\ \scr{D}_3&\coloneq \{(a,b)\in\Bbb{C}\times\Bbb{C}\colon\ a^2=b^2\not\in(-\infty,0)\},\\ \scr{D}_1&\coloneq \scr{D}_2\cup\scr{D}_3. \endaligned $$ \par Here $\widehat{\Bbb{C}}=\Bbb{C}\cup\{\infty\}$. In [J. M. Borwein\ and R. E. Crandall, op. cit.] it was shown that $\scr{D}_1\subset\scr{D}_0$ and it was conjectured that in fact $\scr{D}_1=\scr{D}_0$. In the present paper the authors prove this conjecture. \par Let $|a/b|=1$, $a^2\not=b^2$ and let $p_n/q_n$ denote the $n$-th approximant of $\scr{S}(a,b)$. From standard theory of continued fractions it follows (for even $n$) that $$ \frac{p_n}{q_n}-\frac{p_{n-1}}{q_{n-1}} = - \frac{a^n b^n n!^2}{q_n q_{n-1}}. $$ The authors show that the right side of the above equation does not tend to zero, so that $\scr{S}(a,b)$, and hence $\scr{R}_1(a,b)$, does not converge (in fact, more is shown, in that the odd and even parts of $\scr{S}(a,b)$ tend to distinct limits). This is achieved through careful analysis of the asymptotics of certain infinite matrix products and of certain exponential sums. This analysis is then applied to matrix products derived from modified versions of the usual recurrence relation for $q_n$. \par The results of this analysis appear to have applications that go beyond the divergence results for $\scr{R}_1(a,b)$, and at the end of the paper some generalizations of $\scr{R}_1(a,b)$ are considered.