Colloquium publications.

158

THE BOSTON COLLOQUIUM.

a paper of Jacobi,* published posthumously in 1868. The developments of Jacobi were, however, of a purely numerical nature.
On this side they have been perfected recently by Fr. Meyer [83].
The first example of a functional extension was given by Hermite
in his famous memoir [84] upon the transcendence of e, and the
theory has been developed since independently of each other by
Pincherle and Pade.
To explain the nature of the generalization it will be desirable
first to refer to the mode in which a continued fraction is commonly generated. Two numbers or functions, f and f, are given,
from which a sequence of other numbers or functions is obtained
by placing
f2 = Xlfi -fo
(3) f3= X2f2 —.fl.A = Xf3 -A,
in which the hi are determined in accordance with some stated
law. For the quotient f0.f/, we obtain successively
Ao 1 1
A -f —\1 1 *A- 2 AS
13
and it therefore gives rise to the continued fraction
1 1
(4), _
(2 - X3
By means of the equations (3) each fn+l can be expressed linearly
in terms of the initial quantitiesf0,fl. Thus
(5) fn+l= Al, nIfl + Ao, n+lJo,
in which Ao, +,, Al, n+l are polynomials in the elements Xi. It is
easy to see that these polynomials both satisfy the same difference
* "Allgemeine Theorie der kettenbruchihnlichen Algorithrnen, in welchen
jede Zahl aus drei vorhergehenden gebildet wird." Journ. fir Math., vol. 69
(1868), p. 29.