2.1 Direct Calculation of Padé Approximants
In this section we consider the calculation of the denominators of Padé aDDroximants directly from the explicit determinantal formula (1.1.8) for L^M- 1. We are able to do this principally for the class of functions which may be represented as
Results for functions with the representations and follow as corollaries. In Section 1.2, we showed that explicit calculation of the numerator and denominator for each Padé approximant of the exponential function is possible using the direct method, but existence of a simple explicit form for the numerator polynomial is special to this case. Once the denominator of an [L/M] Padé approximant is constructed, the formula (1.1.9) leads directly to the numerator coefficients; equivalently, we may of order z .
Of course, we do not suggest that numerical calculations are to be made using determinantal formulas, and in Section 2.4 we consider these problems. Nor do we suggest that the method of this section is always the best considei to be defined by truncation of beyond terms
The coefficients of the Maclaurin expansion of (la) are
Substituting in (1.4.8) for L^M- 1 we find that
Reduction of (1.3) is simplified by defining p to be the product of the leading elements of the rows
we need the determinant
To calculate the coefficient of z7 in
following, in which the dotted lines enclose a deleted column:
Sequential row subtraction and expansion about the ( 1,1 ) element leads to
By using this technique recursively.
Making M—j simplifications of the type leading from (1.4) to (1.5), i.e., expansion by top left-hand element after row reduction, we are led to define the product of common factors
Using the same technique of reduction, we obtain
From (1.6), (1.7), and (1.8) there is substantial cancellation, and
completing the derivation of the Pade denominator. We may derive acorollary from this result, by noting that
Consequently the Padé denominators of
are given by
Furthermore, by choosing y=l, we obtain the special case of the exponential function of Section 1.2.
for which explicit expressions for the [L/M] Pade approximants (with L^M— 1) can be given. As in (1.4), we construct the determinant C(L/M) by defining
Likewise, the Padé denominators of the asymptotic expansion of
are given by replacing 2 with yz and letting 00. We find
There is another class of formal series, derived from coefficients
which generate the power series
(1.12)These determinants may be evaluated recursively from the recurrence relation
Hence, using the same methods as for the hypergeometric function (1.2), explicit expressions for the Pade approximants may be obtained. If the coefficients were given by
instead of by (1.11), substitute
If the coefficients were given by
substitute w=Az and let A —> oo. Thus results from series generated by (1.13) or (1.14) are special cases of results derived from (1.12). We pursue the direct calculation no further, because the approach based on the Q.D. algorithm (4.4.17) is algebraically simpler. We refer to Wynn  for explicit formulas using the Q.D. algorithm.