Diagonal continued fraction

A continued fraction

$$ a _ {0} + \frac{b _ {1} \mid }{\mid a _ {1} } + \dots + \frac{b _ {n} \mid }{\mid a _ {n} } + \dots , $$

in which the sequences $ \{ a _ {n} \} _ {n=} 0 ^ \omega $ and $ \{ b _ {n} \} _ {n=} 1 ^ \omega $ must satisfy the following conditions:

1) the numbers $ a _ {n} $ and $ b _ {n} $ are integers; $ | b _ {n} | = 1 $; $ a _ {n} \geq 1 $ if $ n \geq 1 $; $ a _ \omega \geq 2 $ if $ 0 < \omega < \infty $;

2) $ b _ {n} + a _ {n} \geq 1 $ for all $ n $; if $ \omega = \infty $, then $ b _ {n} + a _ {n} \geq 2 $ for an infinite set of indices $ n $;

3) $ Q _ {n} < Q _ {n+} 1 $ for all $ n $;

4) the partial fractions of the continued fraction are all irreducible fractions $ A / B $ such that $ | r - A/B | < 1 / 2B ^ {2} $ and $ B > 0 $, where $ r $ is value of the continued fraction.

For each real number $ r $ there exists one and only one diagonal continued fraction with $ r $ as its value; this fraction is periodic if $ r $ is a quadratic irrationality.

Comments

After truncation and evaluation one obtains

$$ a _ {0} + \frac{b _ {1} \mid }{\mid a _ {1} } + \dots + \frac{b _ {n} \mid }{\mid a _ {n} } = \frac{P _ {n} }{Q _ {n} } , $$

with $ P _ {n} , Q _ {n} \in \mathbf Z $, $ ( P _ {n} , Q _ {n} ) = 1 $, $ Q _ {n} > 0 $. These are the numbers $ Q _ {n} $ alluded to in condition . The continued fraction as described above for a real number $ x _ {0} $ can be obtained by the nearest integer algorithm, that is, $ a _ {0} = \langle x _ {0} \rangle $, $ x _ {1} = 1 / ( x _ {0} - a _ {0} ) $, $ a _ {1} = \langle x _ {1} \rangle $, $ x _ {2} = 1 / ( x _ {1} - a _ {1} ) $, $ a _ {2} = \langle x _ {2} \rangle $, etc., where $ \langle x\rangle $ denotes the nearest integer to $ x $. It is also possible to use the entier function $ [ x] $ instead of $ \langle x\rangle $. One then has the continued fraction algorithm which is more commonly used.

The adjective "diagonal" stems from the fact that $ b _ {n} = \pm 1 $ for all $ n $.

References

[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8
[a2]	O. Perron, "Die Lehre von den Kettenbrüchen" , I , Teubner (1977)