Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 165

In one important respect Padg's investigation has a narrower
reach than Pincherle's and needs completion. The existence of a
second group of associated polynomials-the Pn Q, R of Pincherle-is not brought to light. As has been already pointed out,
it is this second group of polynomials which is the true analogue
of the convergent of an ordinary continued fraction and which
must take precedence in considering the convergence of the algorithm or the closeness of the approximation afforded to the initial
functions. Pincherle's definition of convergence [82] is not, however, so framed as to require explicitly the introduction of these
polynomials. If the given system of difference equations is
(14) J,+3 = cnfT+2 + dn+fnl + n (n = 0, 1, 2...),
the continued fraction is said by him to be convergent when the
two following conditions are fulfilled:
(1) There is a system of integrals F, F', F^ of (14) such that
F /F,, F"/IF have limits for n = o, and these limits are different
from 0.
(2) There is also one particular integral -called by Pincherle
the integrale distinto- the ratio of which to every other integral
of (14) has the limit zero.
Pincherle's interest is evidently concentrated upon this principal integral. It seems to me, however, more natural to call
the algorithm convergent when the ratios Qn/lP and RJIPn (cf.
Equations 12 and 13) converge to finite limits for n = oo. Under
ordinary circumstances these limits will doubtless coincide with
the ratios of the generating functions, fl/f0 andf2/fo.
In the case of an ordinary continued fraction the two definitions
coalesce. For suppose that the nth convergent N,/D of (4') has
the limit L. Then Nn - LD is such an integral of the difference equation,
fn = Xtfn-l + /,nfn-2,
that its ratio to any other integral, k1N + k2D,, has the limit 0.
Conversely, if the principal integral N, - LDn exists, there must
be a limit L for the continued fraction. Possibly the case in