Vtyjkyuk

Padé approximants are a particular type of rational approximation. The $L, M$ Padé approximant is denoted by

$$[L/M] = P_L(x)/Q_M(x)$$

where $P_L(x)$ is a polynomial of degree less than or equal to $L$, and $Q_M(x)$ is a polynomial of degree less than or equal to $M$. The formal power series

$$f(x) = sum _j=0^infty f_jx^j$$

determines the coefficients by the equation

$$f(x) -P_L(x)/Q_M(x)=O(x^L+M+1).$$

There is also an extra condition added by Baker which states that

$$Q_M(0)= 1$$

A proof at the beginning of Essentials of Padé approximants shows that the $[L/M]$ Padé approximant is unique when it exists (Frobenius, 1881).

Now, it is also well known that Jacobi made a breakthrough in 1846. In particular, he applied Cramer's rule to get a determinant solution to the Padé approximant equations.

For $P_L(x)$ and $Q_M(x)$, a simplification of Cauchy's solution to the problem of rational interpolation produces two determinant solution formulas:

The numerator, $P_L(x)$, can be represented by

$$labelten
P_L(x)=mathrmdet,beginvmatrix f_L-M+1 & f_L-M+2 & cdots & f_L+1\ vdots & vdots & ddots & vdots \f_L & f_L+1 & cdots &f_L+M \ sum_j=M^L f_j-Mx^j & sum_j=M-1^L f_j-M+1x^j & cdots & sum_j=0^L f_jx^j endvmatrix $$

Similarly, the denominator can be represented by

$$labelnine
Q_M(x)=mathrmdet,beginvmatrix f_L-M+1 & f_L-M+2 & cdots & f_L+1\ vdots & vdots & ddots & vdots\f_L & f_L+1 & cdots &f_L+M \ x^M & x^M-1 & cdots & 1 endvmatrix $$

However, it isn't clear to me how Jacobi found these determinant solutions. If I go and review the source -

Jacobi, C. G. J. (1846) Uber die Darstellung einer Reihe Gegebner Werthe durch eine Gebrochne Rationale Function. J. Reine Angew. Math. (Crelle) 30, 127-156.

I find that Crelle's journal contains the necessary information on pages 127-156 of the 1846 issue. However, the paper is in German and I cannot read it.

Does anyone know if this paper has been translated into English? Is there some way I could find this out on my own? I am looking to do the derivation myself and wasn't able to find an equivalent article in English.