4.1 Definitions and the Recurrence Relation

In this chapter we do not aspire to summarize the companion volume of Jones and Thron [1980] which is devoted to the general theory of continued fractions. There is a selected bibliography on continued-fraction theory at the end of this volume. Here, we set out to present a working knowledge of the basic concepts of continued fractions, so that we may give a self- contained account of how continued-fraction theory supplements our un­derstanding of Pade approximation. The discovery of continued fractions in the West seems to have been by Bombelli [1572]; Jones and Thron [1980] and Brezinski give historical surveys. In Section 4.7, we quote the basic convergence theorems for general continued fractions, and refer to the companion volume for the proofs. We are primarily concerned with con­tinued fractions associated with power series, for which the continued fractions happen to be Pade approximants. Indeed, in the next chapter we will see that S-fractions are associated with Stieltjes series and that real /-fractions are associated with Hamburger series. The convergents of these fractions form simple sequences in the Pade table.

There is no doubt that part of Pade-approximation theory grew out of continued-fraction theory. We choose to regard the Pade table as the fundamental set of rational approximants, and the convergents of various continued fractions derived from power series as particular subsequences of the Pade table. We suggest that which continued-fraction representation is the most useful is often seen most clearly by considering first which sequence from the Pade table has the desired asymptotic behavior or rate of

The entries in (1.1), a, and 6,, are called the elements of the continued fraction. They are usually real or complex numbers. The fraction may be written more compactly as with precisely the same meaning as (1.1). By truncating the fractions, we define its convergents, which we denote by ratios /!,/#, for / = 0,1,2   We find

 

 

 

 

If the fraction has only a finite number of elements, it takes the form

 

 

 

(1.4) is called a terminating

This is equivalent to (1.2) with fraction. The value of a terminating fraction is defined by finite arithmetic. We also note that (1.4) is the («+ l)st convergent of (1.2). In general, a continued fraction is said to converge and have the value v if exists, it meaning a set of numbers c0, (',, c2,... to be added. The word "series is also used as the value of the sum indicated, provided this value is finite. The same verbal ambiguity arises with continued fractions. Expressions such as (1.1), (1.2), or

The name "series" is used to describe are to be found in the literature. They denote the fact that the pairs (av o,), (a2,b2), (a3, ft3),... define the continued fraction, or else the value of the fraction if it converges. Once noticed, the ambiguity causes no confusion, and is unimportant in practice.

 

Part of the definition (1.5) of convergence of a continued fraction refers to the ratios An/Bn which are the values of the convergents of the continued fraction. It is possible to construct different continued fractions which have all their convergents equal in value. Such fractions are called equivalent, and, by definition, equivalent fractions all have the same value.

By division of the "first" numerator and denominator by ft,, and by division of the th" numerator and denominator by bn for « = 2,3,4,..., the denominator elements have been reduced to unity.

Another simple example shows that the numerator elements may be reduced to unity. We find that

(1.7)

This freedom of representation of the continued fractions using different elements constitutes a group of equivalence transformations. A general member of the group is represented by which are required to be invertin terms ot the parameters ible.

 The convergents of the continued fractions are ratios A,,/Bn, as is emphasized by (1.3) and (1.5). However, it is useful to define the numerators An and denominators Bn separately, but consistently, so that the («+l)th convergent is given by (1.4). The definitions and consistency are ex­pressed by

With this definition, the ratio AH/Bn is the (n+ l)th convergent of (1.9).

Proof. By inspection, A0/B0 and        are the values given by (1.3). We

prove (1.10) and (1.11) by induction. Suppose that they hold for i = m. Then the ( m + 1 )th convergent of ( 1.9) is

 

convergent of (9) from the

 

 

we replace bm

To obtain the

 

by

 

 

defined by (1.10) and (1.11), we find that

wherever it appears in the algebraic expression for

Since b does not occur in the algebraic expressions for where the induction hypothesis has been used to obtain (1.13). Clearly, the definitions (1.10) and (1.11) for i = m+\ are consistent with the values of the (w + 2)th convergent derived in (1.13).

In this theorem we have derived the most important formula needed for continued-fraction theory: the recurrence relation for the numerators A, and the denominators Bi defined by (1.10) and (1.11). As an example, we may inspect A2/B2 in (1.3) and see that it is given correctly by the recurrence formula. The trivial modification of taking bo=0 allows the recurrence to apply to the fraction (1.6). A consequence of Theorem 4.1.1 is that it shows that the following alternative definition of convergence of a continued fraction is entirely equivalent to the previous one.

 

Alternative Definition. The continued fraction

 

 

is said to converge and have value v if the ratio An/Bn of the quantities A and Bn defined recursively by (1.10) and (1.11) tends to v as n^-cc.

 

 

 

 

 

Exercise 2. Show that

 

 

 

 

Exercise 3. Let An/Bn be the n th convergent of

 

 

as described in the text. Prove that the elements are given in terms of the numerators and denominators by bQ=AQ, ax=Ax—          b\—B\ and for

i >2,