Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 159

equation as./,
f,+l = Xf, -.fn —l;
and for their initial values we have
A1 = 1, A0O, = 0,
A1,2-=1 A0,2= —1.
Consequently A,,, and - A,, are the numerator and denominator
of the (n - l)th convergent of (4).
When the generating relations have the form.fo = X.f + CL2f2,
Af= X2f2 + /3f3,
the resultant continued fraction is
(4') +, **
' '- X3- +
A distinction then appears between the system of functions
(A1.,+, - Ao,,+) and the system which consists of the numerator
and denominator of the nth convergent. Though the quotient
of the two functions of either system is the nth convergent, the
former pair of functions satisfy the same relation of recurrence as
the f, namely,
fn = Xn+lfn+l1 'n+2 I +2;
while the corresponding relation for the other system is
gn = Xng,1- + /ign-_
The latter equation is called by Pincherle [77, a] the inverse of
the former. In the continued fraction (4) we took i = - 1 so
that the two relations were coincident.
The immediate generalization of these considerations is obtained
by taking m + 1 initial quantities fO, f, ',,f, in place of two.
With a very slight change of notation we may write