Rationality of Euler-type Generalised Continued Fraction

My question concerns the following:

If we have a convergent series $$S$$ (in some field) equivalent to a Euler-type GCF:

$$S = a_0 + a_0 a_1 + a_0 a_1 a_2 + \cdots = \cfrac{a_0}{1-\cfrac{a_1}{1+a_1 - \cfrac{a_2}{1 + a_2 - ...}}}$$ where $$a \in \mathbb{Q}$$

And $$S \in \mathbb{Q}$$ or $$S \notin \mathbb{Q}$$

And now we take a second series:

$$T = a_0 + a_0 a_1 x + a_0 a_1 a_2 x^2 + \cdots = \cfrac{a_0}{1-\cfrac{a_1 x}{1+a_1 x - \cfrac{a_2x}{1 + a_2x - ...}}}$$

for some $$x \in \mathbb{Q}$$

Is there any relation between the rationality of S and that of T, especially in a p-adic field?

Thank you

• What's given about $a_0,a_1,a_2,\ldots$? And what are CFs doing here at all? Jul 5, 2020 at 15:36
• Its a reference to this en.wikipedia.org/wiki/Euler%27s_continued_fraction_formula. The '$a$'s should be rational numbers. Jul 5, 2020 at 19:19
• The reason for referencing it, is because CFs have a very clear way of indicating rationality, which I suspect might be relevant. Jul 5, 2020 at 19:26

Basically, the question asks whether the rationality of $$f(1)$$ is related to the rationality of $$f(x)$$ for some rational $$x$$, where $$f(x)$$ is any power series in $$x$$ with nonzero rational coefficients and $$f(1)$$ convergent.
Consider $$f(x)=\exp\big(ax(1-x)\big)$$ with $$a\neq 0$$ rational (and $$|a|_p$$ small enough in the $$p$$-adic case). Then $$f(1)=1$$, but $$f(x)$$ is irrational (even transcendental as known since Hermite in the real case and since Mahler in the $$p$$-adic case) for rational $$x\notin\{0,1\}$$. To ensure $$f(x)$$ has nonzero coefficients, it suffices to add a suitable rational function if needed.
This is easily reversed to have $$f(x)$$ rational but not $$f(1)$$ [replace $$f(z)$$ by $$f(xz)$$ and $$x$$ by $$1/x$$].
• How about if the '$a$'s are integer prime powers, and the field is $Q_p$? Jul 6, 2020 at 7:12