Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 139

existence of the three regular types of continued fractions is assured. The necessary and sufficient condition that the table shall
be normal is that no one of the determinants
C a-3+1 Ca-9-+2.. Ca
- a- +2 Ca-P+3 a+ (C, o;
ac -. = 0 ifi < 0)
Ca Ca+l Ca+i-1
shall vanish [16, a, p. 35]. It will be noticed that the determinants are of the same sort as those which play so conspicuous a
r6le in Hadamard's discussion of series representing functions
with polar singularities.
So far as I am aware, the normal character of the table has
been established as yet only in the following cases: (1) for the
exponential series [37] and for (1 + x)m when m is not an integer
[35, d],t by Padg; and (2) for the series of Stieltjes, by myself [45].
The construction of Padg's table leads at once to a number of new
and important questions. The numerators and the denominators of
the approximants constitute groups of polynomials which it is only
natural to expect will be characterized by common or kindred
properties. The table then affords a suitable basis for the classification of polynomials. Thus, for example, the polynomials of
t At least half of the table for F(a, 1, y, x) has a normal character. This was
proved incidentally in my thesis [76] by showing that the remainders corresponding to approximants on or above the diagonal of the table were all distinct.
The method of conformal representation was there employed, but the same fact
can also be demonstrated very simply by means of Gauss' relationes inter contiguas
(formulas (19) and (20) of [34]). The approximants in the other half of my
table (Cf. [76], p. 44) were constructed on different principles from Pade's,
the approximation being made simultaneously with reference to two points,
x = 0 and = oo, but the resulting continued fractions were of the same form
as Pade's. It is noteworthy that the relationes inter contiguas lead to such a
table rather than to the one of Pade's construction.
In the case of F( —m, 1, 1, -x) = (1 + x)m the half of Pad6's table below the
diagonal is also normal, since the reciprocal of the approximants in the lower
half are the approximants in the upper half of the table for
F(m, 1, 1, -x) = (1 - x)-n.
The normal character of the table for ex then follows since e"=lim F(g, 1, 1, x/g).
g = X