The $n$-th convergent is defined as

$$R_n(x) = \frac{P_n(x)}{Q_n(x)}=[1;x,x^2,\cdots,x^n]=1+\frac{1}{x+}\frac{1}{x^2+}\frac{1}{x^3+\cdots}\frac{1}{x^n}$$ where $P_n(x), Q(x)$ are polynomials recursively defined by

$$P_k(x)=x^kP_{k-1}(x)+P_{k-2}(x) \mbox{ with } P_{0}(x)=1, P_{1}(x)=x+1,\\ Q_k(x)=x^kQ_{k-1}(x)+Q_{k-2}(x) \mbox{ with } Q_0(x)=1, Q_1(x)=x.$$

I am trying to find if we can get anything better than what I have obtained empirically so far. I've found some related continued fractions, and like mine, their expansions also involve terms such as $x^{k(k+1)/2}$. See sections 4.2 and 4.3 in this document (also available here in case the link is broken). My interest in $P_k(x)$ and $Q_k(x)$ is that it allows you to easily generate large prime numbers, see here. The focus on large primes will be the object of another question, here I am only interested in asymptotic expansions.

Here are the results that I obtained empirically:

  • $P_k(x) = \mu_k(x)R_\infty(x)\cdot x^{k(k+1)/2}$
  • $Q_k(x) = \eta_k(x)\cdot x^{k(k+1)/2}$
  • $P_k(x)-Q_k(x) = \tau_k(x)\cdot x^{(k-1)(k+2)/2}$

with $ \mu_k(x)-\eta_k(x)\rightarrow 0$ as $k \rightarrow\infty$. Also, $\mu_k(x),\eta_k(x)$ and (to a lesser degree) $\tau_k(x)$ are close to $1$ if $k\geq 1$ and $x>6$. Of course, $\lim_{k\rightarrow\infty} P_k(x)/Q_k(x) = R_\infty(x)$, which is the value of the infinite continued fraction $[1; x,x^2,x^3,\cdots]$.

My question: Getting more precise results for the functions $\mu_k,\eta_k,\tau_k$, for instance their limit (as a function of $x$) as $k\rightarrow\infty$.

Other interesting facts

A Taylor series around $x=0$ is also available for $R_\infty(x)$: the first coefficients obtained with Mathematica here, and correct up to $x^{12}$, are $2, -1, 2, -4, 7, -12, 22, -41, 74, -133, 243, -444, 806$. I did a reverse lookup on that particular sequence (see here) and found another continued fraction that has the exact same Taylor series. So I conjecture that both continued fractions must be identical. Here is the other one:

$$\small 2 - x/(1+2x - x^3/(1+2x^2 - x^5/(1+2x^3 - x^7/(1+2x^4 - x^9/(1+2x^5 - x^{11}/(1 - ...))))))$$

Not sure why that one is listed in OEIS and mine, though much simpler, is not.

Finally, one of my other goals is to obtain an exact, simple closed form, for $\lfloor \log P_k(x)\rfloor$ when $x$ is an integer. Indeed, this is the reason I was interested in that continued fraction in the first place, after obtaining such a close form for the nested radical

$$\sqrt{1+\sqrt{x+\sqrt{x^2+\sqrt{x^3+\cdots}}}} .$$

See here the details for the nested radical.


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