Suppose $b_1, b_2, b_3, \dots \in \Bbb{R}$ satisfy the Riccati-type recurrence $$b_{k+1}=\frac{1+kb_k}{k-b_k},\quad k\ge 1.$$

Is it true that such a sequence reaches infinitely many positive as well as negative values? It appears to be so.


Yes, choose $a_1$ such that $\frac{1}{b_1}=\tan(a_1)$, and let $a_{k+1}=a_k-\arctan\frac{1}{k}$, then we have $\frac{1}{b_k}=\tan(a_k)$, since $\sum_k \arctan\frac{1}{k}=\infty$, the conjecture follows.

  • $\begingroup$ Note that if $b_1=0$ we can take $a_1=\frac{\pi}{2}$ $\endgroup$ – Bonbon May 4 '18 at 16:45
  • $\begingroup$ Not sure if this proves the claim. $\endgroup$ – T. Amdeberhan May 4 '18 at 17:23
  • $\begingroup$ @T.Amdeberhan where makes it "not sure"? You mean $b_i$ may be zero ? If you mean this, its my fault. I should write it as $b_i=cot(a_i)=tan(\frac{\pi}{4}-a_i)$. $\endgroup$ – Bonbon May 4 '18 at 19:19
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    $\begingroup$ It's very nice. If you need more details, the sequence $a_k$ is decreasing to $-\infty$ with jumps going to 0, so its tangent achieves infinitely many change signs. The trick is quite natural: your homographies can be viewed in the projective line (given by rotations), and with a coordinate changes this can be viewed as standard rotations, which after taking these parameters (on the universal covering), yield translations. $\endgroup$ – YCor May 4 '18 at 19:45
  • $\begingroup$ Yes, a great comment! $\endgroup$ – Bonbon May 4 '18 at 19:52

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