4.2 Continued Fractions Derived from Maclaurin Series

A formal power series may be manipulated into the form of a continued fraction very easily. In this section, we ignore all questions of convergence. We assume that all the inverses we need exist and that we do not encounter degenerate cases.

 

We calculate the reciprocal of the series

which allows the reexpansion

 

Next we calculate the reciprocal of the series

 

which allows another reexpansion

The given power series is

 

 

 

 

 

It is clear [Salzer, 1962], that by forming the reciprocal series, we have devised an iterative procedure which allows us to write formally

 

 

which corresponds to the series (2.1). The convergents of (2.4) are rational fractions in the variable z. To be quite general, we assume only that the resultant fraction representing the power series takes the form

 

 

 

and (2.4) is just a special case of (2.5). The first few convergents of (2.5) may be easily calculated:

 

 

 

 

 

We see that (2.6) is equivalent to (1.3) with the replacement at —>atz for all i. Following the analysis of Section 4.1, especially Theorem 4.1.1, we see that the numerators and denominators of (2.5) are generated by

 

 

 

 

(2.7a)

 and

 

 

 

 

(2.7b)

 The connection between the convergents of the fractions (2.4) and (2.5) and the entries in the Padé table is expressed by

 

and

Theorem 4.2.1. Provided that c, and every coefficient c\n are nonzero, the continued fraction (2.4) has the Maclaurin expansion (2.1). In this case, the Padé approximants of {2.1) are identified with the convergents of the continued fraction by

 

 

for M=0,1,2    

Remarks. No statement is implied by this theorem about the domains of convergence in the z-plane, if any, of either the series (2.1) or the fraction (2.4). If nontrivial domains of convergence exist, they are likely to be different. Even if (2.1) and (2.4) are convergent, the theorem does not directly assert equality of these values.

Proof. Since each expression (2.1), (2.2), (2.3) has the same formal Maclaurin expansion, the first part of the theorem is true by induction. Toestablish (2.8), we note that

 

 

 

 

 

 

 

 

 

 We prove (2.8) by induction. Suppose that

 

 

 

 

 

 

Hence the fractions Am(z)/Bm(z) have numerators and denominators of the requisite orders for all m, and power series which agree with (2.1) to order zm inclusive. Consequently the fractions Am(z)/Bm(z) are the Pade ap­proximants of (2.1) of the orders indicated by (2.8).

 

for m = 0,1,2,..., M. Then

 

 

 

 

Notice that the fractions {Am(z)/Bm(z)} defined in this section occupy a descending staircase sequence in the Pade table, which starts with a horizon­tal tread, as shown in Figure 1.

 

This is of the general type

 

As an example of a continued fraction of this type, we may use five terms of the Maclaurin expansion of exp(z) to show that

 

 

This example shows an advantage of using (2.5) rather than (2.4), because the elements of the fraction may be taken to be integers with this represen­tation. We will derive the general term of (2.11) in Section 4.6. If we consider the reciprocal of (2.11) and replace z by — z, we get a different representa­tion:

 

The numerators and denominators of (2.13) are derived from

 

 

 

 

 

 

and

 

 

 

 

 

 

The Padé approximants which are the convergents of (2.13) are given by

 

and

 

These occupy a descending sequence in the Padé table which begins with a stair, as shown in Figure 2.

The comparison of (2.5) or (2.13) with the sequence of Padé approxi­mants indicates that (2.5) is to be preferred in particular asymptotic regions of the z-plane where |/(z)| is increasing, and (2.13) is to be preferred where |/(z)| is decreasing as |z| increases.

 

 

 

If functions are even or odd, they are degenerate in a rather trivial way, and there is no purpose in making a great issue of this. If the function is

 

Figure 2. A descending staircase sequence in the Pade table corresponding to convergents of (2.13).

 

 

 

 

It is customary to adopt the most convenient form of continued fraction without discussing alternative possible representations. Consideration of the sequence of convergents as they appear in the Padé table with the desired asymptotic properties and consideration of any known degeneracies give a guide to the best continued-fraction representation to use.

Exercise 1. Prove that the Padé approximants which are the convergents of (2.13) are given by (2.15).

even, either one uses

 

or

If the given function is odd, normally one chooses

 

Exercise 2. Which entries in the Padé table are occupied by the conver­gents of (2.16)?