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

(1.2)

Substituting in (1.4.8) for L^M- 1 we find that

(1.3)

Reduction of (1.3) is simplified by defining p to be the product of the leading elements of the rows

 

 

so that

 

(1.4)

 

 

 

 

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

 

 

 

(1.5)

 

By using this technique recursively.

(1.6)

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

 

(1.7)

and then

 

Using the same technique of reduction, we obtain

 

 

 

 

(1.8)

From (1.6), (1.7), and (1.8) there is substantial cancellation, and

 

 

 

 

 

Hence, for

 

 

 

 

(1.9)

 

 

 

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

 

 

 

 

(1.10)

 Furthermore, by choosing y=l, we obtain the special case of the ex­ponential function of Section 1.2.

 

 

 

 

where

 

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

 

 

(1.11)

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

 

and let

(1.13)

 

 

instead of by (1.11), substitute

If the coefficients were given by

 (1.14)

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 [1967] for explicit formulas using the Q.D. algorithm.