25 
and hence by (15) 
yp uv “ae 
ON [ep (t) |= @) | ze | AEN 1) 
T'(@+a) (t—1)% 
Not only ¢, (#), but also the function 
_ top, (l 
has the property that the operation J applied to it gives a coefficient- 
function satisfying the condition (7), the inequality (14) for / being 
left unaltered. We only have in this case y= -—a, instead of 
y =O, and the domain of validity is determined by A (#) > — a, 
or, according to (18), by 
te Ween tects cote Road Te, 0 (20) 
where 
Fee ese tic ne at, (Oe eee (A) 
That is: the domain of validity of (7) is the domain of absolute 
convergence of the series (1) (for d, is arbitrarily small). 
For the whole right-hand member of (19), that is for w(«) we 
therefore have the inequality 
| w (a) |< M | (@ + by at-C+4)|. 2 2 . … (22) 
where /, now, satisfies the condition 
eager eee walt taan geent Nes) 
If, at last, the characteristic 2' of g(t) is greater than —1 or 
equal to — 1, then, after Pincarrre, the coefficientfunction can be 
expressed in terms of that of another generating function p, (0), 
with a characteristic less than — 1. First, let 
Ly areal 
then PINcHERLE considers the additional function 
p (¢) 
t) = — D-1| —— }}, 
p, @) D (25) 
having a characteristic 4’—1, which, therefore, is less than —4, 
so that the corresponding coefticient-function w, (#) satisfies, in the 
domain (re) >a—1-+4d,, the inequality 
| w, (#) |<< M| (@ + db)! ax—C—-144) | 
with 
hee i tod, od 
The coefficientfunction w (xv) of p(t) is connected with the latter 
by the formula *) 
w (cz) = A [(z — 1) w, (# — 1)] 
1) PINCHERLE, |. c., p. 64. 
