4.7 Convergence of Continued Fractions

The derivation given in Section 4.6 of the continued-fraction expansions of the familiar functions of mathematics is purely algebraic. We can justify the usual meaning of the equality signs in Section 4.6 if we can show that each continued fraction converges to a limit function, and

this limit function is the same function as the one from which the fraction originated.

 

 

 

We simply state the relevant theorems here, and refer to the companion volume of Jones and Thron for the proofs.Since each convergent of a continued fraction of the types discussed in Section 4.6 is a rational function of the variable z, one expects to be able to specify conditions which are sufficient to ensure that the limit function is meromorphic in z. Normally, most authors prove theorems by showing that the convergents of a continued fraction [such as (6.14)] converge on bounded domains of the z-plane which do not contain the poles or other singularities, if any, of the limit function. However, some authors prefer to discuss convergence of continued fractions in terms of the chordal metric, which means the same as convergence on the Riemann sphere. This ambiguity rarely leads to confusion, provided the trap is anticipated. We use conver­gence in the ordinary sense, unless otherwise stated.

We consider a continued fraction in the reduced standard form

 

 

Naturally enough, convergence criteria for continued fractions are always essentially properties of the "tail" of the fraction; the value of b0 is totally immaterial. The divergence condition is an important criterion for the convergence of (7.1).

Theorem 4.7.1. If the continued fraction (7.1) converges for any nonzero value of z, then either must diverge.

Remarks. Note that divergence of one of the series (7.2) is a necessary but not a sufficient condition for convergence of (7.1). It must be supplemented by further conditions on the elements a, if convergence is to be proved, as is done in Theorem 4.7.3.

where />, >0 for all i. The fraction (7.3) converges if and only if , bl diverges.

We do not prove Theorem 4.7.1. Instead we show the scope of such proofs by proving a similar result due to Seidel which is self-contained and very relevant to Stieltjes series.

Theorem 4.7.2. Consider the continued fraction

 

Proof. We first prove that convergence of (7.3) implies divergence of First we prove an identity which shows that the sequence of convergents of (7.3) may be written as an alternating series.

Since

The alternating-series test states [Ferrar, 1938, p. 47] that if (i) un is a decreasing seauence, (ii) u„ —»0 as n—* oo, and (iii) u„ >0 for all n, then the

is called an alternating series and it converges.

series

 

 

We show that the condition that 1 b diverges implies that

Note that

 

 

 

 and this shows that

 

 

Hence condition (ii) of the alternating-series test is valid, and we deduce

 

that

If 1bn is not divergent, let

 

 

 We may easily prove by induction that

 

 

 using (7.5b), and hence

 

 

 

 

Consequently, the terms of (7.6) do not decrease in modulus, and so the ratios An/Bn do not converge to any limiting value.

An interesting application of Theorem 4.7.2 is that we may show directly that the Stieltjes fraction (5.5.24) converges on the positive real z-axis. The connection between Theorems 4.7.1 and 4.7.2 becomes evident from an equivalence transformation, as we state in Exercise 1.

Our next theorem provides a sufficient condition for the convergence of a continued fraction.

Theorem 4.7.3 (Parabola theorem) [Scott and Wall, 1940b; Thron, 1974]. The continued fraction

 

 

converges provided that

(i) a may be found in the range-it/2 < a < 77/2, and n0 exists such that the elements of (7.7) satisfy

 

 

jor alt n>n0\ and

(ii) the divergence condition (7.2) is satisfied.

Corollary. If it so happens that the sequence {an} has a nonzero limit, let an —>a. We choose to take a=0. The key condition (7.8) is then satisfied if we impose a constraint on z, namely

 

This defines a parabolic domain 9 shown in Fig. 1. The parabola has its focus at z = 0 and its axis running through z= — 1/(4a). The geometric

 

Figure 1. The parabolic domain \'P of Theorem 4.7.3 corresponding to a = ir/\2.

 

interpretation of (7.8) is that z must be nearer the origin than the directrix of the parabola. The hypothesis that an—>a, a^0, is also sufficient to satisfy the divergence condition, and so this extra hypothesis allows a simple corollary of the parabola theorem.

 

 

and

Theorem 4.7.4 (Cardioid theorem) [Paydon and Wall, 1942; Dennis and Wall, 1945; Thron, 1974]. Provided that n 0 and k may be found such that

(c) the divergence condition (7.2) is satisfied, then the fraction (7.7) con- erges for all z in the cardioid

 

 

(7.9)

Figure 2. The parabolic domain '.'P, of Theorem 4.7.4 corresponding to k = 1.07.

 

Interpretation. Condition (b) of the theorem requires that all the partial numerator coefficients an of the continued fraction lie in a parabolic domain shown in Figure 2. If the conditions (a), (b), and (c) of the theorem are satisfied, convergence of the fraction is assured in the cardioid shown in Figure 3. Note that a larger value of k gives a more restrictive parabolic constraint and a larger cardioid domain of convergence for the fraction.

Theorem 4.7.5. If the sequence of convergents of (7.7) converges uniformly in \z\<R with R> 0, to a limit function f(z), then f(z) is analytic in | z | and its power series generates the fraction (7.7).

 

then

converges to a meromorphic function of z. 1

(7.7) converges to a function f(z) which is meromorphic in the cut z-plane. The

Remark. This theorem is a consequence of Theorem 4.7.3. and is proved using Weierstrass's theorem [Titchmarsh, 1939, p. 95; Copson, 1948, p. 97].

 

the fraction (7.7)

 

Theorem 4.7.6 [Van Vleck, 1904].

Figure 3. The cardioid domain of Theorem 4.7.4 corresponding to k = 1.07.

 

cut is placed in the shadow of ~(4a)~x from the origin, as shown in Figure 4. In each case, convergence is uniform on any compact set containing no poles of the limit function, and the limit function has the continued fraction expansion (7.7).

Next, we will use some of these theorems to prove the quoted results of the previous section. Van Vleck's theorem is used to prove (6.13), (6.14) and (6.16); (6.15) is proved by using the cardioid theorem.

Proof of (6.13). We have shown that both left- and right-hand side of the equation

with

 

have the same formal power-series expan-

sion. Since an—>0, Theorem 4.7.6 states that (7.10) is an identity for all z not a zero of 0Fx(a; z).

Figure 4. The cut from z0 = — (4a) 1 to oo in the complex z-plane for Theorem 4.7.6

 

Proof of (6.14). We have shown that both left- and right-hand side of the equation

 

 

with

 

 

and

 

 

 

 

have the same formal power-series expansion. Since an —> 0, Theorem 4.7.6 asserts that the right-hand side of (7.11) is convergent and that (7.11) is an identity for all z not a zero of ^F^a, b\ z).

 

 

 

Partial proof of (6.15). We have shown that both left- and right-hand side of the identitywith

 

 

 

 

 

 

 have the same formal expansion. By noting that the ratio of successive numerator coefficients ak/ak+x^ 1, we may show that the divergence condition is satisfied, which is one necessary condition for the convergence of (7.12). It is more convenient to use the variable z'~ —2, so that (7.12) becomes formally

 

 

This equation has the status of a formal algebraic identity, and we seek tc show that it represents an identity between function values in the cu 2'-plane.

 

The sequence an is shown in Figure 5. We see that for any k^Q n0=n0(k) exists such that

 

 

 

 

Figure 5. The numerator elements a„ in the complex plane defined by (7.12).

Hence the continued fraction (7.13) converges for all z' in the cardioid

 

 

Since this is true for any k>0, the continued fraction converges for all r except on — oo<z'<0, which corresponds to the positive real z = axis.

To establish equality between left- and right-hand side of (7.13), we note that for the special case of b=0, a>0, we have a strict Stieltjes series in the variable z', and the theory of Section 5.5 is applicable. Using Carleman's

To

theorem, (7.13) is established directly as a true equality for

 

 

Theorem 4.7.6 asserts that

have the same formal expansion. Since

(7.14) is an identity valid in the z-plane cut along

 

This is, of

extend this argument to the cases in question, we refer to Wall [1945].

Proof of (6.16). We have shown that both left- and right-hand side of the equation

 

 

with

 

 

and

 

 

course, the usual domain of definition of a 2FX hypergeometric function. Hence the fraction (7.14) converges for all z not on the cut and not a zero of

 

 

Exercise 1. Consider the fraction

 

 

with b, >0 for all i. Show that the divergence condition (7.2) is precisely the

 

diverges.

condition that

Exercise 2. Prove that the condition that cient to satisfy the divergence condition. Exercise 3. Prove that the fraction

 

is suffi-

 

 

 

 

converges for  1 and diverges for v> 1.