18
$\begingroup$

Simplified repost of Are these continued fractions of integrals known? on MSE

EDIT: The period of the oscillations of $$f(s)=\dfrac1{1+\dfrac s{1+\dfrac{s^2/2!}{1+\dfrac{s^3/3!}{1+\cdots}}}}$$ appears to converge to $$\dfrac4e=1.4715\cdots$$ as @user42355 commented. Can this be proven?

Empirically, the argmins are

-2.12897, -3.73409, -5.26940, -6.78494, -8.29119, -9.79168, -11.2881, -12.7814, -14.2723,
-15.7613, -17.2487, -18.7348, -20.2198, -21.7038, -23.1870, -24.6695, -26.1513, -27.6326,
-29.1133, -30.5937, -32.0736, -33.5531, -35.0323, -36.5111, -37.9897, -39.4680, -40.9461,
-42.4239, -43.9015, -45.3790, -46.8562, -48.3332, -49.8101, -51.2869, -52.7635, -54.2399,
-55.7162, -57.1924, -58.6685, -60.1445, -61.6203, -63.0961, -64.5717, -66.0473, -67.5228,
...

with consecutive differences (note insufficient precision beyond fourth decimal)

         1.60512, 1.53531, 1.51554, 1.50625, 1.50049, 1.49642, 1.49330, 1.49090, 
1.48900, 1.48740, 1.48610, 1.48500, 1.48400, 1.48320, 1.48250, 1.48180, 1.48130, 
1.48070, 1.48040, 1.47990, 1.47950, 1.47920, 1.47880, 1.47860, 1.47830, 1.47810, 
1.47780, 1.47760, 1.47750, 1.47720, 1.47700, 1.47690, 1.47680, 1.47660, 1.47640, 
1.47630, 1.47620, 1.47610, 1.47600, 1.47580, 1.47580, 1.47560, 1.47560, 1.47550, ...

Define the continued fraction integral transform $$\{\mathcal If(t)\}(s)=\dfrac{f(s)}{1+\dfrac{\int_0^s f(x)\,dx}{1+\dfrac{\int_0^s\int_0^u f(x)\,dx\,du}{1+\cdots}}}$$ where $f\in C^\infty(\Bbb R)$. Has this been studied?

Elementary properties include $\{\mathcal I(0)\}(s)=0$, $\{\mathcal If\}(0)=f(0)$ and $1+\{\mathcal If\}(s)=f'(s)/\{\mathcal If'\}(s)$ for all non-constant $f$ such that $f(0)=0$.

When $f(t)\equiv1$, the function $\{\mathcal I(1)\}(s)$, mentioned in A319173 with no further information, has the following graph

enter image description here

Zooming out (orange with 140 terms; blue with 100 terms; magenta with 79 terms), we observe excellent convergence, until the plots branch off to either 0 or 1 depending on the parity of the number of terms.

enter image description here

How do we explain the regular oscillating behaviour over the negative reals?

Improve this question
$\endgroup$
3
  • $\begingroup$ The almost regular oscillating behavior over the negative reals is clear to see, but what about the monotonic behavior over the positive reals? How fast does it decay to zero? $\endgroup$
    – Somos
    Commented Jun 8, 2023 at 21:28
  • 1
    $\begingroup$ It is not hard to prove that the function is meromorphic. The plots are a bit misleading: the function has (probably infinitely many) negative poles, and the oscillatory behavior is also present for $s>0$, only the amplitudes are smaller, so the function is not monotone decreasing for $s>0$, because of the oscillations. The period of the oscillations seems to be $\frac{4}{e}$. I think this should be related to $\frac{s^n}{n!}$ becoming approximately $1$ at around $n \approx e s$, so the fractional part of $e s$ should have some control over the value. $\endgroup$
    – user42355
    Commented Jun 15, 2023 at 7:57
  • $\begingroup$ Are there any reasons to define such a transformation? $\endgroup$
    – Alexey Ustinov
    Commented Nov 23, 2023 at 13:22

0

Reset to default

You must log in to answer this question.