4.5.2 General Fractions for Degenerate Cases
Next we consider the situation when Viskovatov's method breaks down and the series cannot be represented as a regular C-fraction. The algorithm leaves us with a representation such as
a correct procedure is to allow a numerator
equal to cy'z and continue to develop the fraction. This procedure defines the general corresponding fraction, or general C-fraction [Leighton and Scott, 1939; Scott and Wall, 1940a, b], represented by
with at ^0 and a, > 1 for all i. Such fractions need not occupy a simple staircase sequence in the Padé table. Consider the series
The third convergent of (5.12) corresponding to this function is
whereas the [3/3] Padé approximant of /(z) is
which is the [3/3] Padé approximant precisely.Many examples of functions having natural boundaries are expressed as series expansions 2°L0c,z' with Hadamard gaps, such as (2.2.4). We say that the expansion has a gap («,, n ■) if Cj—0 for all j in nt The series is
said to have Hadamard gaps if, for some A>0, the series has an infinity of gaps («,, n\) such that nt/n\ 1 +A. In this context, an interesting example of a general C-fraction is Ramanujan's fraction
which has a natural boundary on |z| = 1 although its power series does not have the Hadamard gaps.
The orderly relation between continued fractions and the Pade table is restored by P-fractions [Magnus, 1962]. P stands for principal part plus, and all convergents of P-fractions lie in the Pade table. Using the variable
a P-fraction is
where each b/u) is a polynomial in w of degree Nt precisely. For example, the third convergent of the P-fraction of f(z) given by (5.13) is
This is the [3/3] Padé approximant of f(z). In fact the recurrence relations (4.4) for A, and P, and the accuracy-through-order conditions verify that the /th convergent of (5.15) is a diagonal \L,/L,\ Padé approximant, where
Further, by replacing c0 in (5.15) with />0(w), another polynomial, P- fractions of functions f(z) having a Laurent series with a finite principal part are directly defined. Magnus defined P-fractions in this way so thai every entry in the Pade table may be uniquely associated with a P-fraction of :"[(:) for some integer .v.