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