314 
which, by iteration, passes into 
ll, [*—"g (5] 
FP (¢—r) 
It is easy to see that the latter equality is also valid for negative 
integral values of r. It is, however, remarkable that the same equality 
holds for not-integral values of r. This property I have made use 
of in the investigation of a function represented’) by a binomial series 
140 (Q)=(-y 
x—l 
ze ) the most typical series in which a coefficientfunction 
n 
can be expanded. The object of this note is to give a proof of the 
general validity of the equality in question. 
2. We substitute —a for r and replace g(t) by the expression 
p(t): (—1), in order to have always to deal with functions which 
are regular for ¢=o. In accordance with Rigmann’s *) definition 
of the derivative of negative order —a of a function we assume as 
such the following one 
ol) te 
(—1) D-=- — = B 
(¢—1)+ (a) (uw — 1) 
t 
In this we take as path of integration the half-line beginning at 
u=t, whose prolonged part passes through vw == 1; and we assign 
the same arguments, lying between add and +2—4d, to 
u—l and w—t. Then the so-defined derivative is also regular 
without the circle (1,1) and zero for t=; and by the substitution 
u—1=(t—1):s we find after a slight reduction the expansion 
(Ie De pd _ a Du +Vgn 
(t— 1) met Dn +1 4+ e)(t—1)4! be 
0 
(6) 
which, therefore, is related in a simple manner to the expansion 
for p(t itself. The order of this derivative, as will appear imme- 
diately from this series, is « less than that of (dé), and therefore 
negative together with the latter order. On this supposition we 
may apply the operation / to it in the form (1), so that by this 
the evistence of the first member of the equation to be proved, ‘viz. 
Tye ande (7) 
(le) Mete) [ID] 0 
1) These Proceedings Vol. XXII, N°. 1. Nieuw Archief v. Wisk. XIII, 2e stuk (1920). 
*) See BoreL, Lecons sur les séries a termes positifs, p. 74. The constant a 
occurring there is taken equal to here, in connection with the regularity of 
g(t) for t=. 
