1.2 Pade Approximants to the Exponential Function

The coefficients c, of the Maclaurin expansion of the exponential func­tion are sufficiently simple that explicit forms of the numerator and denominator of the Pade approximants can be found. In this section we will calculate the denominator

 

The numerator follows by an extremely simple and elegant trick, based on the identity and this derivation is discussed in Section 1.5. Pade, in his thesis, elaborated the properties of his approximants with special emphasis on the example of the exponential function: it is a beautiful example of how the approximants work in an ideal situation. Further properties of Pade approximants of exp(z) are to be found in Sections 4.6, 5.7, II.3.3 and 3.4.

Our task is to calculate

 

 

 

It is easier to begin with the constant term in (2.1), and so we define

(2.1)

 

which is the coefficient of the "1" in the lowerright-hand corner of (2.1),

 

 

We assume that

 

If this condition does not hold, the factorial

functions must be suitably reinterpreted as gamma functions for the analy­sis to be valid. We remove the denominators from each row, by defining and then

 

 

(2-3)

In (2.3), the determinant has M rows. Subtract the (M— l)th row from the A/th, then the (A/-2)th row from the (M- l)th, etc. The identity in column is used repeatedly. In column 1 of (2.3),

M — 2; etc., and so one finds that This is a (M-l)X(M-l) determinant with a form identical to (2.3) but with M replaced by M— 1. Consequently, an obvious inductive argument shows that

Thus, for the case and for the case The sign pattern of (2.6) distinguishes Polyâ frequency series, to which we refer in Section 5.7. The row operations we have performed to deduce (2.6) from (2.2) are still permissible with the form (1), except that the situation is which more complicated. We consider the coefficient of is

 

 

 

 

 where the column surrounded by analysis: define

: is deleted. We perform a similar

 

 

 

(2.8)

and thenSubtracting rows, and using the identity (2.4),

 

 

 

 

(2.8). We make j similar reductions from (2.8) to obtain

which again is an

determinant with a form similar to Removing a common factor from each row, The analysis now follows the familiar pattern using the identity (2.4), and we deduce that (2.9)

 

The sign of the right-hand side of (2.9) is easily determined to be the same as that of (2.6), because the determinants (2.2) and (2.7) have the same dimension, and are expanded by the same top right-hand elements recur­sively. Hence

(2.10)

Notice that (2.6) emerges as the special case with j=0. Consequently we have and

Following the method of Section 1.6, we may deduce from (2.11) that and hence the [L/M] Padé approximant for exp(z) is