4.4 Various Representations of Continued Fractions
We are primarily concerned with continued fractions whose convergents form a sequence of Padé approximants. We will consider continued fractions containing the complex variable z explicitly, so as to maintain the connection with Padé approximation. The formulas remain valid with z = 1 ; this may be more useful for other purposes connected with the general theory of continued fractions. We will first establish the existence of tridiagonal determinantal formulas for the numerators and denominators of a continued fraction.
Theorem 4.4.1. The convergents of the continued fraction
have numerators given, for n = 0,1,2,..., by
and denominators given, for n = 1,2,3,..., by
Proof. We expand the determinant (4.2) by its last column. This leads to the equation
which is the standard recurrence relation (2.7a).
tridiagonal representation (4.2) is valid for the numerators of the convergents of the continued fraction (4.1).
We deduce from (2.14b) that the tridiagonal representation (4.3) is valid for the denominators of the convergents of the continued fraction (4.2).
From (2.14a) we deduce that the
Theorem 4.4.2. The convergents of the continued fraction
and denominators given by (4.3).
Proof The method of proof is identical to that of Theorem 4.4.1.
have numerators, given for n — 2,3,..., by
The determinants (4.2) and (4.3) express the numerators and denominators of Padé approximants of types [M/M} or [M+ 1 /M], according to the discussion of Section 4.2. Equations (4.2) and (4.3) involve the elements ai and bl of the continued fraction (4.1). Next, we reexpress (4.1) in terms of Hankel determinants, so as to relate (4.1), (4.2), and (4.3) more directly to the coefficients c■ of the Maclaurin expansion of f(z).
Theorem 4.4.3. The continued fraction (4.1) may be expressed as
The initial elements of (4.1) are
and for M =2,3,4,..., by
With these values (4.8)—(4.12) for the elements of the continued fraction f(z). the determinantal formulas (4.2) and (4.3) satisfy for all M
where we assume B0(z)= 1.
Remark. The elements ah bt given in (4.8)-(4.12) are only determined up to equivalence transforms if one only requires that (4.1) should have a given Maclaurin expansion; they are determined uniquely by the conditions (4.13).
Proof. In (4.3), we set z = 0 and obtain
Thus we can arrange for the conventional normalization to hold, namely
by choosing the coefficients bt according to (4.10) and (4.12). By inspection, we verify that the initial values (4.8) are chosen so as to satisfy
To establish (4.9) and (4.11) for the elements ajt we use Frobenius identities (3.5.3) and (3.5.7).
1 satisfy the same recurrences (4.17),
as also do the numera-
(4.19) as the denominators
tors A„, (4.5).
The recurrence relation (2.7b) for the even-order denominator
By comparing this equation with (4.17), we see that the definition (4.11) of
The odd-order denominator write as
is also generated by (2.7b), which we
By comparing this equation with (4.19), we see that the definition (4.9) of
provided that all the lower-order denominators are identical. By combining (4.18), (4.20), and (4.8), the representation (4.7) is proved by induction. The
Hence (4.13) is established by induction, using the initial conditions expressed by (4.16).
Theorem 4.4.4. A power series and a continued fraction may be formally identified by
Proof. The method of proof is identical to that of Theorem 4.4.3.
Corollary. Using an equivalence transformation, we may reexpress (4.21) as
and for A/=2,3,4,...,
As we see in (5.5.25) and (5.6.33), such a representation and its contraction are of especial importance to Stieltjes series.
The numerators and denominators generated by recurrence are identical to those of Theorem 4.4.2.
In fact, (4.23) is a formal identity, and it represents the connection between a formal power series and a sequence of [M/M} and [M— 1 /M]Padé approximants. The elements a't of (4.23) are relatively easily calculated directly from the coefficients c, using the Q.D. algorithm; no-one would contemplate using (4.25) for iterative numerical computations in view of the likely ill-conditioning of the Hankel determinants. In the context of numerical computation, it is more usual to write (4.23) as
and for M = 2,3,4,...,
More generally, we use the expansion
Equation (4.26) is a special case of (4.29) with 7=0. With our usual convention that L=M+J, it follows from (4.28) that for M=2,3,4,..., and
Theorem 4.4.5. The elements of (4.29) satisfy, for J5*0 and M= 1,2,3,...,
First proof. Substituting from (4.30),
for A/ = 2,3,4,..., and 73=0. The case for M= 1 is easily verified explicitly, and one may also treat it using the convention that C(L/0)=1. Likewise one verifies (4.32):
Second proof. We consider the algebraic identity from two sequential continued fraction expansions, namely
We make a contraction, given by (5.3) and explained in the next section, on these fractions, and it follows that
By identifying the coefficients in these expansions, (4.31) and (4.32) are proved.
The q.d. algorithm. This algorithm may be used to construct the coefficients ef in the continued-fraction representation (4.26) of the Padé approximants of a given power series. In this context, the initializing values are taken to be as is required by (4.34). Equations (4.31) and (4.32) constitute the body of the algorithm. The elements are normally exhibited in the Q.D. table as shown in Table 1.
Table 1. The Q.D. Table
Its two left columns are specified by (4.35), and the remaining elements are determined by (4.31) and (4.32). The elements along any diagonal are the elements of a continued fraction (4.29).
[The q.d. Table of exp(z)]. The left-hand column consists of zeros, and the next column contains entries
(see Table 2). The rules (4.31) and (4.32) connect entries at the vertices of rhombi in the Q.D. table, enabling Table 2 to be completed. We deduce thatTable 2. Part of the Q.D. Table of exp(r)
which agrees with (6.1a).
One may extend the Q.D. table above its diagonal by defining =0 for /=1,2,3,.... In this way all the elements of the top row except <7° are defined to be zero, and the rhombus rules allow completion of the table. If the coefficients of the reciprocal series are used by setting e'i+] =c'i+]/c', for / = 0,1,2,..., in the notation of (1.6.10), and the appropriate order of calculation is followed, then the whole table may be calculated more stabh by the progressive form of the Q.D. algorithm described in Section 3.6: otherwise the Q.D. algorithm is notoriously unstable. If f(z) is meromor- phic, the "<7" columns converge to the reciprocals of the poles of f(z). provided the moduli of the poles are distinct. Similarly the "e" rows converge to the reciprocals of the zeros of f(z), provided they have distinct moduli. This property follows from (4.28) and (3.7.6) et seq. It is instructive to compare this form of the Q.D. algorithm with that described in Section 3.6: they are not identical.
Throughout this section we have assumed that all the inverses we need do in fact exist. It is quite possible that bk= 0 for some value of k in (4.1) and so only the first k convergents are defined; convergence of the fraction is meaningless. Likewise the Q.D. algorithm may break down if a zero divisor is encountered. Worse, from a numerical point of view, is the possibility that the computed results are generated by rounding error in a zero or near-zero entry. We refer to Claessens and Wuytack  for an extension of (4.31) and (4.32) to the case of a nonnormal Q.D. table.
Exercise 1. Prove Theorem 4.4.2. Exercise 2. Prove Theorem 4.4.4.
Exercise 3. Construct the Q.D. table corresponding to the Maclaurin series coefficients of the function 2^o(a> 1;*).4.5 Different Types of Continued Fractions