Q: What is this notation $[b_0 b_1,…]_\alpha$ with continued fractions in the image below? Especially what does subscript $]_\alpha$ mean?
What I have tried:
I know the notation
$$[b_0 b_1,…]=b_0+\frac{1}{b_1+}\frac{1}{b_2+\ldots }$$ for a simple cfrac. I tried to google translate the text but couldn't make any sense of it. So after translatng the text
Lause 4.11.Olkoon훼∈ℝ∖ℚ, 훼 >0annettu ja olkoon[푏0;푏1,…]훼(4.109)Ketjumurtoalgoritmilla muodostettu lukuun훼liittyvä yksinkertainen ketju-murtolukukehitelmä. Tällöin
I get
Theorem 4.11.Let 훼 ∈ℝ ∖ ℚ, 훼> 0 be given and let [푏 0; 푏 1, …] 훼 (4.109) A simple chain-fractional development related to the number 훼 formed by a chain fracture algorithm. Here
I couldn't understand what this means. I hope someone who knows Finnish language can answer this question, or anybody who has encountered the notation. I haven't seen this kind of notation anywhere else. Thanks.
Best Answer
I'm just going to do my best to make a translation from the Google-English:
Thm 4.11: Let $\alpha \in R \setminus Q, \alpha > 0$ be given, and let $$ [b_0, b_1, \ldots]_\alpha $$ be the simple continued-fraction expression produced by the continued-fraction algorithm. Then $$ \alpha = [b_0, b_1, \ldots]_\alpha. $$
In short: the continued fraction algorithm (presumably described earlier in the paper) produces an expression which, when evaluated, gives $\alpha$. More precisely, the partial terms of the CF form a sequence which has a limit that is $\alpha$.
In really short: the continued fraction algorithm works to produce continued fractions as expected.
The proof(?) that follows appears to say 'suppose that you had two different cfracs for $\alpha$, and that the first difference is in the $m$th term. Then (by various bits of algebra) you'd find (some algebraic relation between the two differing $m$th terms).