Nice sign-expansions of special surreal numbers

(This answer has been substantially edited since its original version.)

1.

The surreal number with sign-expansion $+-^{\omega}++-^{\omega}+++-^{\omega}++++-^{\omega}\cdots$ (indexed by $\omega^2$) can be rewritten in a tidy form, which shows that it does lie in the field generated by $\omega$.

As you read off that sign-expansion, the number follows the following pattern: $$1,1/2,1/4,1/8,\ldots$$ $$+-^{\omega}=\frac1\omega$$ $$\frac2\omega,\frac3\omega,\frac2\omega+\frac1{2\omega},\frac2\omega+\frac1{4\omega},\ldots$$ $$+-^{\omega}++-^{\omega}=\frac2\omega+\frac1{\omega^2}$$ $$+-^{\omega}++-^{\omega}+++-^{\omega}=\frac2\omega+\frac3{\omega^2}+\frac1{\omega^3}$$

So that we end up with $$\frac2\omega+\frac3{\omega^2}+\frac4{\omega^3}+\cdots=\frac{2\omega-1}{\left(\omega-1\right)^2}$$ Now, the fact that the finite sums and the subtler fact that the "infinite sum" works out in this way is not obvious.

2.

$\dfrac{2\omega-1}{\left(\omega-1\right)^2}$ is not actually a ratio of ordinals.

Treating the expression as a rational function of a real variable $\omega$, it diverges at $\omega=1$, but if it could be written as a rational function with positive coefficients (say by multiplying the numerator and denominator by a suitable polynomial), it could not diverge for any positive $\omega$. Incidentally, the inverse of this sort of argument is in Meissner's "Über positive Darstellungen von Polynomen" which contains the theorem that a polynomial which is positive for nonnegative real $x$ (such as $x^2-x+1$) can always be written as a rational function with positive coefficients (such as $(x^3+1)/(x+1)$).

3.

A sign expansion with the finite-orbit property need not be in the field generated by the ordinals.

$+^\omega-^{\omega^2}$ has the finite orbit property (indeed, the only nontrivial tails have the forms $+^\omega-^{\omega^2}$, $-^{\omega^2}$), but $+^\omega-^{\omega^2}=\sqrt{\omega}$ is not in the field generated by the ordinals (and hence certainly not a ratio of ordinals).

4.

A ratio of ordinals that are not both finite can still sometimes have the finite-orbit property. In particular, $\dfrac{1}{\omega}=+-^{\omega}$, and more interestingly, $\dfrac{1}{\omega+1}=\left(+-^{\omega}-+^{\omega}\right)^{\omega}$. The latter expansion has the finite orbit property, since the only forms of tails are $\left(+-^{\omega}-+^{\omega}\right)^{\omega}$, $-^{\omega}-+^{\omega}\left(+-^{\omega}-+^{\omega}\right)^{\omega}$, $-+^{\omega}\left(+-^{\omega}-+^{\omega}\right)^{\omega}$, and $+^{\omega}\left(+-^{\omega}-+^{\omega}\right)^{\omega}$.

$+-^{\omega}-=\dfrac{1}{\omega}-\dfrac{1}{\omega2}=\dfrac{1}{\omega2}$ Then for finite $n$, we have $+-^{\omega}-+^{n}=\dfrac{1}{\omega}-\dfrac{1}{\omega2}+\dfrac{1}{\omega4}+\cdots+\dfrac{1}{\omega2^{n-1}}=\dfrac{1}{\omega}-\dfrac{1}{\omega2^{n+1}}$. We end up with $+-^{\omega}-+^{\omega}=\dfrac{1}{\omega}-\dfrac{1}{\omega^2}$. Analogously, $\left(+-^{\omega}-+^{\omega}\right)^n=\displaystyle\sum_{k=1}^{2n}\dfrac{(-1)^{k-1}}{\omega^k}$. By the expansion of $\dfrac{1}{x+1}$ at infinity, we have $\dfrac{1}{\omega+1}=\left(+-^{\omega}-+^{\omega}\right)^\omega$.

5.

However, a ratio of ordinals need not have the finite-orbit property. In particular, $\dfrac{1}{\omega+2}=+-^{\omega}-+^{\omega}-^1+-^{\omega}+^3-+^\omega-^7+-^\omega+^{15}\cdots$. In a similar way to the analyses of $\dfrac{2\omega-1}{(\omega-1)^2}$ and $\dfrac{1}{\omega+1}$, this follows from the expansion at infinity: $\dfrac{1}{\omega+2}=\displaystyle\sum_{k=1}^{\infty}\dfrac{(-2)^{k-1}}{\omega^k}$.

Enough tools for justifying the above facts about sign expansions above are covered in VIII.2 and VIII.3 from Aaron N. Siegel's book Combinatorial Game Theory, but you can very likely also extract what you need from Philip Ehrlich's "Conway names, the simplicity hierarchy and the surreal number tree".

While I do not know if there is a

"right" surreal generalization of the fact that a real number $r$ is rational if and only if its sign-expansion is eventually periodic

it is perhaps worth noting that Conway has identified what he believes "are perhaps the closest analogue in No of the ordinary rational numbers" (ONAG, p.47). These surreal "fractional numbers" are the continued fractions that terminate at a finite stage. Instead of using the integer part to generate the continued fraction, one uses the omnific integer part. Conway observes that: "If $x$ is fractional, so are $x+1$, $-x$ amd $1/x$ (if $x \neq 0$), but neither the sum nor the product of two fractional numbers need be fractional….” Moreover, it is easy to show that there are fractional numbers in Conway’s sense having sign-expansions that are not eventually periodic.