Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 123

then E G(x) is a series which converges uniformly in any region
71=0
lying, with its boundary, entirely in the interior of the star. The
series may also converge outside the star. Borel * has shown,
furthermore, that the series of Mittag-Leffler is not the only possible one, but there is an infinity of polynomial series sharing the
same property within the star.
It will be noticed that the construction of the series of MittagLeffler is in no wise dependent upon the convergence of the initial
power series. In certain cases, at least, the polynomial series converges when the given series (1) is itself divergent. It is natural
therefore to look for a theory of divergent series based upon convergent series of polynomials. As yet, however, no such theory
has been invented. One of the chief difficulties in the way is that
the polynomial series do not afford a unique mode of representing
an analytic function. Now the difference between any two series
of polynomials for the same function in an assigned area is a third
series which vanishes at every point of the area, though the separate terms do not. This is a decidedly awkward point, and
occasions difficulty in proving or disproving the identity of two
functions expressed by polynomial series. It is true, indeed, that
this difficulty will scarcely present itself when we start with a convergent power series which is to be continued analytically, the
polynomial series then giving continuations of a common function.
But when the series (1) is divergent and there is no known function which it represents, it is an open question whether the different series of polynomials which are obtained from (1) by application of diverse laws will furnish the same or different functions.
If different functions, is there any ground for preferring one series
of polynomials to another?
Up to the present time two essentially different principles seem
to have been followed in the formation of series of polynomials.
In the work of Runge, Borel, Painleve and Mittag-Legfler the coefficients in the polynomials vary with the character of the ana*Ann. de l'Ec. Nor., ser. 2, vol. 16 (1899), p. 132, or Les Series divergentes,
p. 171.