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Consider a sequence of polynomials

$$ \Phi_0(x)=x $$ $$ \Phi_{k+1}(x)=x+\Phi_k(x)^2\quad\quad(k\ge0). $$

It is easy to see that the equation $\Phi_k(x)=1$ has exactly one positive root. Let us denote it by $\xi_k$. I am interested in the asymptotics of this root, as exact as possible. It is not difficult to see that $\Phi_k(\frac14) < \frac12$ for all $k$, whence $\frac14 < \xi_k$. On the other hand, I am able to get an upper bound of the form $\xi_k < \frac14+\frac{c}k$ for some positive constant $c$. It is sufficient for my purposes, but numerical data suggest that $\xi_k$ is close to $\frac14+\frac{\pi^2}{k^2}$ as $k\to\infty$.

Is it possible to prove it? Or, perhaps, such a sequence of polynomials has already been discussed somewhere?

An additional question for reference: consider a power series for the function $$ \frac1{1-\Phi_k(x)} $$

in a neighborhood of zero. Its nth coefficient is equivalent to $C\xi_k^{-n}$ for some $C > 0$, which can be deduced from the standard theory of linear recurrent equations. However, I think that such statements should follow from some well-known theorems. Does anyone know any useful links (preferably in English)?

P.S. It is worth noting that for large $k$, the first terms of the power series are Catalan numbers, but after them there is a long “tail” with smaller coefficients.

QLimbo
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  • It seems to me that your question is very related to this post https://math.stackexchange.com/questions/2063137/series-related-to-the-mandelbrot-set – NN2 Oct 28 '22 at 21:21
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    and perhaps this paper may help https://mapletransactions.org/index.php/maple/article/view/14037/11474 – NN2 Oct 28 '22 at 21:36
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    Looking for very large values of $k$ I think that $\xi_k=\frac 14+\sum_{n=1}^p \frac {a_n}{k^n}$ is more than reasonable but we cannot ignore the role of $a_1$ – Claude Leibovici Oct 29 '22 at 06:08
  • @NN2 Thanks for the link. I'll take a look - I think it's relevant to my topic. – QLimbo Oct 29 '22 at 16:57
  • @ClaudeLeibovici the improvement of the estimate leads only to the fact that some set of cardinality $2^{2^{7000}}$ turns into $2^{2^{200}}$, which has only an abstract meaning. But it is still interesting to know how the sequence behaves. I forgot to ask what the numbers $p$ and $a_n$ mean? – QLimbo Oct 30 '22 at 10:24
  • Think about a polynomial regression in $1/k^n$ using $p$ terms – Claude Leibovici Oct 30 '22 at 10:27
  • @NN2 by the way, the number $π$ appears there when evaluating the roots of polynomials, and the article in which this is established is quite recent. This is not exactly what I want, but it is quite possible that for the root of the equation $Φ_k(x)=1$ it was also obtained by someone. – QLimbo Oct 30 '22 at 14:28
  • I computed the roots and I stay with my previous comment.. – Claude Leibovici Oct 30 '22 at 15:31

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