Continued Fractions, Convergence Theory. Vol. 1, 2nd Editions. Loretzen, Waadeland. Atlantis Press. 2008

4 Chapter 1: Introductory examples
Quite similarly we can construct, from any sequence {b n } of complex numbers, a
continued fraction
∞
Kn=1
1
=
b n
b 1 +
1
1
1
b 2 +
b 3 + ···
= 1 b 1 +
1
b 2 +
1
b 3 + ···, (1.1.7)
or from two sequences, {a n } and {b n } of complex numbers, where all a n ≠0,a
continued fraction
∞
a n
Kn=1
b n
=
b 1 +
a 1
a 2
b 2 + a 3
b 3 + ···
= a 1
b 1 +
a 2
b 2 +
a 3
b 3 + ···. (1.1.8)
We write K(1/b n ) and K(a n /b n ) for these structures.
In all the three cases the nth approximant f n is what we get by truncating the
continued fraction after n fraction terms a k /b k , and convergence means convergence
of {f n }. (1.1.5) and (1.1.7) are obviously special cases of (1.1.8). In the particular
case when in (1.1.7) all b n are natural numbers, we get the regular continued fraction,
well known in number theory.
Let us take a look at the common pattern in the three cases: series, products and
continued fractions (and other constructions for that matter). In all three cases
the construction can be described in the following way: We have a sequence {φ k }
of mappings from Ĉ to Ĉ. By composition we construct a new sequence {Φ n} of
mappings
Φ 1 := φ 1 , Φ n := Φ n−1 ◦ φ n = φ 1 ◦ φ 2 ◦···◦φ n . (1.1.9)
For series we have
φ k (w) :=w + a k ,
and
Φ n (w) =φ 1 ◦ φ 2 ◦···◦φ n (w) =a 1 + a 2 + ···+ a n + w.
For products we have
φ k (w) :=w · a k ,
and
Φ n (w) =φ 1 ◦ φ 2 ◦···◦φ n (w) =a 1 · a 2 ···a n · w.
For continued fractions (1.1.8) we have
and
φ k (w) :=
a k
b k + w ,
Φ n (w) =φ 1 ◦ φ 2 ◦···◦φ n (w) = a 1
b 1 + b 2 +···+ b n + w . (1.1.10)
a 2
a n