5 Padé approximants

5.1 Obtaining Padé approximants

The same paradigm shift can be applied to polynomials : Algorithm 6 that computes the Bezout's coefficients of polynomials, can be rewritten to focus on the convergents of a rational fraction and thereafter rewritten to obtain approximation of non fractional objects. The key problem is to find the appropriate substitute for the Euclidean quotient.

Our start point will be no more where , but where and is a meromorphic function near . The idea is to combine the to obtain a sequence of increasing order in the variable . Therefore, we have to define , the substitute of , by the leading term of . That ``quotient'' will have a negative order i.e., tends to infinity as and the elimination can better be written as , leading to a sequence of remainder whose orders are increasing.

FIG. : Padé approximants for .

Figure 8 applies this process to the expansion of near . The expansion can be written in two flavours, according to your meaning of ``without division''. You can request only integers everywhere and therefore numerators have both an integer part and a litteral part, or you can request only litterals on numerators and therefore have rationals for the remaining coefficients. We obtain the expansions:

.

and the convergents:

These rational fractions are alternatively the and the Padé approximants of , the -Padé being the best fractional approximant of a given function when both and are required.

A nice formula :

5.2 Quotients in arithmetical progression

When applying this algorithm up to order to function , the results obtained can be summarized by Eq. 7, where the first expansion is the raw one and the second is the expansion ``without divisions''.

In other words, the partial quotients stand in arithmetical progression. To obtain a full understanding of this striking feature, the Bessel's functions are required. These functions are define by:

These functions are relevant to our problem because of formulae :

The key point is the recurrence relation:

The foundamental reason for this relation is from differential équations, but a blind proof is nevertheless easy to obtain. We have:

Defining as the quotient of two consecutive Bessel's functions, i.e. , Eq. 8 can be rewritten as:

And the conclusion follows since this constant is nothing but the limit value of as , as obtained by the quotient of the leading term of each factor. Changing into leads to an arthmetical progression of reason , while changing enables to change the first term of the progression. For example:

while