4.5.1 Regular Fractions for Nondegenerate Cases
The type of continued fractions which are fundamental to the representation of power series are the regular C-fractions. These have the form
with a^0 for /=1,2,3,.... They may be constructed from a given power series by Viskovatov's method, the Q.D. algorithm, or any other convenient method. They are called C-fractions because they form the given power series, and the regularity condition is that a, 0 for all /. An iterative reexpansion of the convergents of (5.1) shows that the successive convergents correspond to the [0/0],[1/0],[1/1],[2/1],[2/2],..., sequence of Padé approximants to the given power series as shown in Section 4.2.
If, during the construction of (5.1) from the power series as in (2.1)-(2.4), an at is found to be zero, a different representation, such as the general C-fraction (5.11), must be used.
An alternative form of the regular C-fraction has the representation
with different elements a: from those in (1); we still require that at ^0 for all /, for regularity. Equation (5.2) corresponds to the [0/0],[0/1],
[1/1],[1 /2],[2/2] sequence of Padé approximants.
The simple algebraic identity
leads to a contraction of the continued fraction. By taking we may contract (5.1) and generate its associated fractior
The convergents of A(z) are alternate convergents of C(z), and occupy the diagonal of the Padê table in this case. A particular case of the regular C-fraction is the Stieltjes or S-fraction, which is
witha(>0, /'=1,2,3,... . The properties of the convergents and convergence of S-fractions are discussed extensively in the next chapter in the context of Padé approximation of Stieltjes functions. We will see that if the S-fraction converges (e.g. if the divergence condition of Section 4.7 is satisfied), then
where <f>(?) is a bounded and nondecreasing function defined on
Using the variable w = z , (5.6) is frequently expressed in the form
which is generated by a simple equivalence transformation, and the theor\ of Sections 5.5 and 5.6 shows that
Continued fractions of the type
in which kj =/=0, / =1,2,3, are called J-fractions. If k,> 0 and /, are real
for /'=1,2,3 then (5.9) is called a real J-fraction [Wall, 1931, 1932a,b.
1948]. Such a fraction may be derived from (5.7) by the contraction formula (5.3), with the identifications
for / = 2,3,4,....
Thus we see that the convergents of (5.9) correspond to alternate convergents of (5.7). We will see in Section 5.6 that any convergent real /-fraction has a representation
where «Hw) is a bounded and nondecreasing function defined on ~oo<u< oo. Equation (5.8) is a special case of (5.10) when \b(u) is constant on