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A note on extending Euler’s connection between continued
fractions and power series
John Gill∗
Department of Mathematics, University of Southern Colorado, Pueblo, CO 81001-4901, USA
Received 12 January 1999
Abstract
Euler’s Connection describes an exact equivalence between certain continued fractions and power series. If the partial numerators and denominators of the continued fractions are perturbed slightly, the continued fractions equal power series plus easily computed error terms. These continued fractions may be integrated by the series with another easily computed error term. c1999 Elsevier Science B.V. All rights reserved.
MSC:40A15; 30B70
Keywords:Continued fraction; Power series; Fixed point
1. Introduction
Euler, in 1748, described an exact equivalence between a series and a continued fraction (CF) (see e.g. [1]). In terms of power series (PS) with non-zero coecients this connection may be written
c0+
c1 1 −
c2
c1
1 +c2
c1
− · · ·
−
cn
cn−1
1 + cn
cn−1
≡c0+c1+· · ·+cnn (1)
for each n. Or (1) may be expressed in an equivalent form: 1
1−
a1 1 +b1−
a2 1 +b2−
· · · −
an
1 +bn
= 1 +a1+a1a22+· · ·+a1a2· · ·ann; (1a)
where an=bn=cn+1=cn.
∗E-mail address: [email protected] (J. Gill)
If the partial numerators and denominators of the CF in (1a) are perturbed slightly, so thatan 6=bn,
the nth approximants of the new CF no longer represent exactly the nth sum of a nite PS. It is possible, however, to express the nth approximants as the nth partial sum of a particular power series (the “connecting” series) plus a convenient error term:
1 1−
a1 1 +b1−
a2 1 +b2−
· · · −
an
1 +bn
= 1 +e1+e22+· · ·+enn+En(): (1b)
The formula derived in this paper generates the relationship in (1b) in such a manner that, when |an−bn|is small and a simple convergence criterion is met by the CF the expressionE= limn→∞En is eciently bounded: |E()|6C||for values of the variable in a disk about the origin. Surpris-ingly, the formula derives from consideration of the complex dynamic behavior of linear fractional transformations (LFTs). As →0, each |an−bn| →0 and (1b) tends to (1a). Thus, it is possible
to actually observe the continuous coalescence of two distinct arithmetic expansions, a CF and a PS, into a single unied entity.
In the inequality above, C is an easily computed constant, and the value of depends upon the distribution of the set of repelling xed points of the LFTs that generate the CF.
If the connecting series is integrated, one obtains the approximate integral of the CF, with an error
|Eint()|6C||2:
2. Development of the theory
Our immediate goal is to express the special CF and PS in (1a) in terms of attracting and
repelling xed points of LFTs. Then the relationship will be extended in a very natural manner in this dynamical setting.
Divide both sides of (1a) by a complex number 6= 0: 1
−
a1
a1+ 1−
a2
a2+ 1− · · ·
−
an
an+ 1
= 1
+ a1
+ a1a2
2+· · ·+a1a2· · ·an
n−1
: (2)
Next, set n:=an for n= 1;2; : : : ; so that (2) becomes
1
− 1
1
+ 1− 2
2
+ 1−
· · · −
n
n
+ 1
=1
+ 1
2+
12
3
2+· · ·+12· · ·n
n+1
n
which, through an equivalence transformation, may be written as
1
−
1
1+−
2
2+− · · ·
−
n
n+
= 1
+ 1
2+
12
3
2+· · ·+ 12· · ·n
n+1
n: (3)
The CF on the left contains a special T-fraction [1], and {n} and are xed points of certain
LFTs, as we shall see a little further along.
Next, we will generalize the CF in (3) as follows:
1
1 −
11
1+1−
22
2+2 −
which reduces to (3) when n≡. The sequences {n} and {n} connect the CF to the dynamic
behavior of LFTs, described briey in the following background exposition: The periodic CF
a b+
a b+ · · ·
in most instances converges to the attracting xed point of the generating LFT:
f(w) = a
b+w: (4)
To see this is the case, assume there are two distinct xed points of (4), and , with unequal moduli. One can write (4) as f(w) ==(+−w), which can then be expressed in an implicit form appropriate for iteration:
f(w)− f(w)−=K
w−
w−; K=
with ||¡||: (5)
Here is the attracting xed point of f, and is its repelling xed point. Upon iteration, (5) becomes
fn(w)−
fn(w)−=K nw−
w−:
Clearly fn(w)→ as n→ ∞ if w6=. Therefore,
a b+
a b+
a
b+· · ·=
+−
+−
+−· · ·=:
Returning now to the more general setting, we rst isolate the regular CF structure imbedded in (3a):
11
1+1 −
22
2+2 −
· · · (6)
The nth approximant of (6) may be expressed as Fn(0) =f1 ◦f2 ◦ · · · ◦fn(0), where fn(w) =
nn=(n+n−w). n and n are the xed points of fn. Thus, we may write
1
1−Fn(w)
= 1
1−
11
1+1−
22
2+2 − · · ·
−
nn
n+n−w
: (7)
Although (7) with w= 0, as a generalization of the CF in (3), no longer equals the nth partial sum of a PS, we shall now expand it in a quasi-series form.
Theorem 1. IfFn(0) =f1◦f2◦ · · · ◦fn(0);with fn(w) =nn=(n+n−w);
1
1−Fn(w)
=
n
X
m=1
1
m m−1
Y
j=1
j
j
!
m−1
+
1
n+1−w
n
Y
j=1
j
j
!
n+E
where
En=− n
X
m=1
m
Y
j=1
j
j
!
mm(Fmn+1(w))
; m(t) =
m−m+1 (m−t)(m+1−t)
;
and
Fmn+1(w) =fm+1◦fm+2◦ · · · ◦fn(w); Fnn+1(w) =w; 0 Y
j=1
j
j
!
:=1:
Proof.
From (5), with Kn:=n=n,
fn(w)−n
fn(w)−n
=Kn
w−n
w−n
⇒ fn(w)−n+n−n
fn(w)−n
=Kn
w−n+n−n
w−n
⇒ · · · ⇒ 1
fn(w)−n
=−1
n
+ Kn
w−n
=−1
n
+Kn
1
w−n+1
+n(w)
⇒ 1
fn(w)−n
=−1
n
+ Kn
w−n+1
+Knn(w): (9)
Repeated application of (9) leads by induction to (8).
Corollary 1.1. Ifn ≡;then
1
−
1
1+− · · ·
−
n
n+
= 1
−Fn(0)
=
n+1 X
m=1
1
m
m−1
Y
j=1
j
m
−1
;
which is consistent with Euler’s connection as written in(3).
We must establish conditions on {n} and {n} that imply convergence of the CF in (8) (with
w= 0) and produce a small and easily estimated En. If no such conditions are stipulated the CF
and its connecting PS may go their own separate ways.
Example 1. Let n≡1 and n=n, then the CF in (8) is
1 1−
+ 1−
2
+ 2−· · ·= 1 1−
if ||61, and the PS is
1 + 2!+
2
3! +· · ·= e−1
Theorem 2. Suppose there exist positive constants AandB such that|n|6Aand|n|¿B ¿1for
The classical Pringsheim Convergence Criterion [1] will apply if we can show that |n +
n|¿|nn|+ 1. Write From (8) of Theorem 1, we have
and since the Pringsheim Criterion applies to each Fn
m+1(0), giving |Fmn+1(0)|¡1. Thus
Corollary 2.1. Ifn ≡ and the hypotheses of Theorem2 are satised;then
Z z
0:002% the value of the integrated series. Both series have an actual radius of convergence R= 2, and the CF may converge as a meromorphic function well beyond the limiting value of M.
Reference
