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CHAPTER 1
THE CONTINUED FRACTION AS A PRODUCT OF LINEAR FRACTIONAL TRANSFORMATIONS
1. Definitions and Formulas. Let
[MATHEMATICAL EXPRESSION OMITTED]
be an infinite sequence of linear transformations of the variable w into the variable τ, and consider the product τ0τ1 ... τn(w) of the first n + 1 of these transformations, given by
[MATHEMATICAL EXPRESSION OMITTED]
If we write
[MATHEMATICAL EXPRESSION OMITTED]
then the required product is
[MATHEMATICAL EXPRESSION OMITTED]
If we now put w = ∞ and then let n tend to ∞, the resulting infinite expression which is generated is called a continued fraction.
In case at most a finite number of the quantities τ0τ1 ... τn(∞) are meaningless, and the limit
[MATHEMATICAL EXPRESSION OMITTED]
exists and is finite, then the continued fraction is said to converge, and v is called its value. Thus, the value of a continued fraction is the limit of an infinite sequence of images, under certain transformations, of a fixed point w = ∞.
A glance at the above expression for τ0τ1 ... τn(w) will show immediately that the transformations τp(w) may as well be replaced by the simpler transformations
[MATHEMATICAL EXPRESSION OMITTED] (1.1)
We observe that t0t1 ... tn(0) = t0t1 ... tn+1(∞). The continued fraction which is generated is
[MATHEMATICAL EXPRESSION OMITTED] (1.2)
and the value of the continued fraction is
[MATHEMATICAL EXPRESSION OMITTED]
We shall introduce some definitions with a view toward making these ideas more precise. The numbers ap and bp, called elements, may be any complex numbers; ap/bp is called the pth partial quotient, ap is the pth partial numerator, and bp is the pth partial denominator. The quantity
[MATHEMATICAL EXPRESSION OMITTED]
is called the nth. approximant. The 0-th approximant is t0(0) = b0. We shall exhibit some properties of the approximants.
By mathematical induction it is readily shown that
[MATHEMATICAL EXPRESSION OMITTED] (1.3)
where the quantities An-1An, Bn-1, Bn are independent of w and may be computed by means of the fundamental recurrence formulas:
[MATHEMATICAL EXPRESSION OMITTED] (1.4)
In fact, this may be verified directly for n = 0. If true for n = k, then
[MATHEMATICAL EXPRESSION OMITTED]
so that the statement is true for n = k + 1 and therefore for all n.
We call An the nth numerator and Bn the nth denominator. The nth approximant is given by
t0t1 ... tn(0) = An/Bn.
The determinant of the transformation t = t0t1 ... tn(w) is
[MATHEMATICAL EXPRESSION OMITTED]
so that
[MATHEMATICAL EXPRESSION OMITTED], (1.5)
where a0 must be taken equal to unity. The formula (1.5) is called the determinant formula.
We are now prepared to make the following definition.
Definition 1.1. The continued fraction (1.2) is said to converge or to be convergent if at most a finite number of its denominators Bp vanish, and if the limit of its sequence of approximants
[MATHEMATICAL EXPRESSION OMITTED] (1.6)
exists and is finite. Otherwise, the continued fraction is said to diverge or to be divergent. The value of a continued fraction is defined to be the limit (1.6) of its sequence of approximants. No value is assigned to a divergent continued fraction.
We remark that if the partial numerators ap are all different from zero so that, by (1.5), An and Bn cannot both vanish, then the existence of the finite limit (1.6) insures that but a finite number of the denominators Bn can vanish. Hence, in this important case, the continued fraction converges if (and only if) the limit (1.6) exists and is finite.
Frequently, the elements ap and bp of the continued fraction depend upon one or more parameters, or may themselves be regarded as independent variables. In such cases, one is naturally concerned with the question of uniform convergence. We make the following definition.
Definition 1.2. If the elements ap and bp of a continued fraction are functions of one or more variables over a certain domain D, then the continued fraction is said to converge uniformly over D if it converges for all values of the variable or variables in D, and if its sequence of approximants converges uniformly over D.
The first part of the book is concerned largely with the problem of determining conditions upon the elements ap and bp of the continued fraction which are sufficient to insure convergence. This convergence problem is essentially more complex and interesting than the corresponding problem for infinite series.
We have adopted the natural notation for a continued fraction. Other notations in more or less common use are
[MATHEMATICAL EXPRESSION OMITTED]
and
[MATHEMATICAL EXPRESSION OMITTED]
2. Continued Fractions and Series. The following theorem establishes a connection between certain continued fractions and infinite series.
Theorem 2.1. If the denominators Bp of the continued fraction
[MATHEMATICAL EXPRESSION OMITTED] (2.1)
are all different from zero> and if we put
[MATHEMATICAL EXPRESSION OMITTED] (2.2)
then the continued fraction (2.1) is equivalent to the continued fraction
[MATHEMATICAL EXPRESSION OMITTED] (2.3)
in the sense that the nth approximants of (2.1) and (2.3) are equal to one another for all values of n. Moreover, for arbitrary numbers ρp, the nth numerator of (2.3) is equal to the sum of the first n terms of the infinite series
[MATHEMATICAL EXPRESSION OMITTED] (2.4)
and the nth denominator is equal to unity.
Proof. The sum of the first n terms of the infinite series
[MATHEMATICAL EXPRESSION OMITTED] (2.5)
is An/Bn, the nth approximant of (2.1). By the determinant formula (1.5), this infinite series may be written as
[MATHEMATICAL EXPRESSION OMITTED]
which, by (2.2), is the series (2.4). Now, the linear transformation s = sp(w) = 1 + ρpw may be written in the form
[MATHEMATICAL EXPRESSION OMITTED]
If we apply the first n of these in succession, and then put w = 0, we obtain as the product, on the one hand, the sum of the first n terms of the series (2.4), and, on the other hand, the nth approximant of the continued fraction (2.3). Consequently, the nth . approximants of (2.1) and (2.3) are equal to the sum of the first n terms of the series (2.4), and hence to each other, for all values of n. One may readily verify by means of the fundamental recurrence formulas that the nth denominator of (2.3) is unity, and therefore the nth numerator is equal to the sum of the first n terms of (2.4).
This completes the proof of Theorem 2.1.
We note for future reference that if bp = 1, p = 2, 3, 4, ..., then we have the formulas
[MATHEMATICAL EXPRESSION OMITTED] (2.6)
where ρ0 must be taken equal to zero; and
[MATHEMATICAL EXPRESSION OMITTED] (2.7)
where we must take a1 = 0, ρ-1 = ρ0 = 0. These may be readily verified by means of (2.2) and (1.4).
3. Equivalence Transformations. It is often convenient to throw the continued fraction (1.2) into another form by means of a so-called equivalence transformation. This consists in multiplying numerators and denominators of successive fractions by numbers different from zero:
[MATHEMATICAL EXPRESSION OMITTED] (3.1)
One may easily show by mathematical induction that this continued fraction has precisely the approximants of (1.2). In fact, the pth numerator and denominator of (3.1) are
c1c2 ... cpAp and c1c2 ... cpBp,
respectively, where Ap and Bp are the pth numerator and denominator of (1.2). This can be readily verified by means of the fundamental recurrence formulas (1.4).
If, conversely, two continued fractions with nonvanishing partial numerators have a common sequence of approximants, then either can be transformed into the other by means of an equivalence transformation. In fact, if Ap' and Bp' are the pth numerator and denominator of one continued fraction, and Ap and Bp are those of the other, then there must exist constants Cp ≠ 0 such that
[MATHEMATICAL EXPRESSION OMITTED] (3.2)
Let
[MATHEMATICAL EXPRESSION OMITTED]
Then, since Ap-1Bp-2 - Ap-2Bp-1 ≠ 0, by virtue of (1.5), we conclude that the elements ap and bp are uniquely determined by the Ap and Bp. Similarly, the elements of the other continued fraction are uniquely determined by the Ap' and Bp'.
Let
[MATHEMATICAL EXPRESSION OMITTED]
so that, by (3.2),
[MATHEMATICAL EXPRESSION OMITTED]
with a like relation for the Bn. Here we must take C-1 = C0 = 1. Consequently, by the preceding, we must have
[MATHEMATICAL EXPRESSION OMITTED]
where cp = Cp/Cp-1. Thus, the two continued fractions are the same up to an equivalence transformation [89].
We note the following important special cases. If bp ≠ 0, p = 1, 2, 3, ..., and we take cp = 1/bp, then (3.1) takes a form in which all the partial denominators are equal to unity. Likewise, if ap ≠ 0, p = 1, 2, 3, ..., and we take c0 = 1, and determine the other cp recurrently by the equations cp-1cpap = 1, p = 1, 2, 3, ..., then (3.1) takes a form in which the partial numerators are all equal to unity.
4. Even and Odd Parts of a Continued Fraction. By the even part of a continued fraction we shall understand the continued fraction whose sequence of approximants is the sequence of even approximants of the given continued fraction. Similarly, the odd part of a continued fraction is the continued fraction whose sequence of approximants is the sequence of odd approximants of the given continued fraction. For the sake of simplicity, we shall write these down for the continued fraction
[MATHEMATICAL EXPRESSION OMITTED] (4.1)
rather than for (1.2). The even part of (4.1) is
[MATHEMATICAL EXPRESSION OMITTED] (4.2)
and the odd part is
[MATHEMATICAL EXPRESSION OMITTED] (4.3)
The even and odd parts of (1.2) may be obtained from (4.2) and by multiplying them by a1 adding b0, and then replacing a1 by a1/b1, and ap by ap/bp-1bn, p = 2, 3, 4, ....
To prove that (4.2) is the even part of (4.1), let us regard (4.1) as being generated by the sequence of transformations
[MATHEMATICAL EXPRESSION OMITTED]
so that t1t2 ... tn(1) = An/Bn, the nth approximant. Let sp(w) = tptp+1(w), p = 1, 2, 3, .... Then
[MATHEMATICAL EXPRESSION OMITTED]
and s1s2 ... sp(1) = t1t2 ... t2p(1) = A2p/B2p, the 2pth approximant of (4.1). Since
[MATHEMATICAL EXPRESSION OMITTED]
it therefore follows that (4.2) is the even part of (4.1). The proof that (4.3) is the odd part of (4.1) can be made in an analogous way.
More general "contraction formulas" and also "extension formulas" will be found in Perron [69, pp. 197-205]. Next to the equivalence transformation, the transformations represented by and (4.3) are perhaps the most useful continued fraction transformations.
Exercise 1
1.1. Let An/Bn, n = 0, 1, 2, ..., be the sequence of approximants of the continued fraction
[MATHEMATICAL EXPRESSION OMITTED]
where ap ≠ 0, a2p-1 ≠ - 1, p = 1, 2, 3, .... Form a continued fraction
[MATHEMATICAL EXPRESSION OMITTED]
having the sequence of approximants [MATHEMATICAL EXPRESSION OMITTED]. [58, 124.]
1.2. Let c be any number such that the denominators Bp(z) of the continued fraction
[MATHEMATICAL EXPRESSION OMITTED]
are all different from zero for z = -c. Put
[MATHEMATICAL EXPRESSION OMITTED]
Then, the continued fraction may be written in the form
[MATHEMATICAL EXPRESSION OMITTED]
where ζ = z/c.
1.3. Show how to transform the continued fraction of 1.2 into the continued fraction
[MATHEMATICAL EXPRESSION OMITTED]
1.4. Let u and v be the two roots of the quadratic equation
[chi square] - bx - a = 0, (a ≠0, b ≠ 0),
and suppose that | u | [greater than or equal to0] | v |. Show that the nth approximant of the periodic continued fraction
[MATHEMATICAL EXPRESSION OMITTED]
can be written in the form
[MATHEMATICAL EXPRESSION OMITTED]
and hence deduce the facts concerning convergence and divergence of the continued fraction.
Suggestion. Since a = - uv, b = u + v, the continued fraction may be written in the form
[MATHEMATICAL EXPRESSION OMITTED]
Now apply Theorem 2.1, after making a suitable equivalence transformation.
1.5. Show that
[MATHEMATICAL EXPRESSION OMITTED]
1.6. Show that if ap ≠ 0 is real for p ≥ n, then all the approximants of the continued fraction (4.1) from and after the nth approximant lie upon a circle (or straight line) in the complex plane.
1.7. Let u0, u1, u2, ... be numbers all except possibly the first different from zero, and put Un = u0 + u1 + ... + un. Let
[MATHEMATICAL EXPRESSION OMITTED]
Show that the 2nth and (2n + 1)th approximants of the continued fraction
[MATHEMATICAL EXPRESSION OMITTED]
are Un and Un + un+1/2, respectively [58].
(Continues…)
Excerpted from "Analytic Theory of Continued Fractions"
by .
Copyright © 2018 Hubert Stanley Wall.
Excerpted by permission of Dover Publications, Inc..
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