24 
values of a'), if for the derivative of any negative order the defini- 
tion of Riemann is adopted, which in the present case, a neigh- 
bourhood of infinity being regarded, can be expressed by the identity 
en p(t) (w—t)*—! 
(—1)«D Gie aan nit fj plu)du. 0; <6) 
Since as domain of ¢ and w the part of the plane outside a 
certain circle with centre (1) is considered, it will be convenient to 
assume for path of integration between w=? and w= o the half- 
line which has the same direction as the vector from u=1 to 
u==t. The quantities u —t and w—1 then have the same argu- 
ments and (w— t)*:(w—1)* is real. With these agreements we 
have the expansion 
DD 
Be 
en EN l +4) (¢—1)»+1 om © (17) 
so that the derivative of eam order — « of the expression 
y (t): (¢ — 1)* is, as q(t) itself, regular and zero at infinity. The 
characteristic of the derivative is, however, a less and this makes 
it possible, by means of (15), to express the coefficientfunction w («) 
of a generating function p(t) with characteristic 4’< —1 in terms 
of another generating function g,(t) whose characteristic is any small 
amount less than —1. The function g, (¢) is constructed in such a 
way that the given function p(t) is the derivative of a certain 
negative order —a of p‚(t):(t —1)* and the number «a is selected 
from an aggregate of positive values, whose upper limit is equal 
to the difference between A and — 1. In other words, if ¢ (¢) is 
given by (3), we take 
——~ en 
9, (1) = Si Det 
0 
where the meaning of c’, is given by 
awk F (n+ 1) Cn 
en == TAD 
with 
gesl Ad 8 ate eee) 
J, being any small positive number. Then, according to (17) 
soy ane ee 
1) | have communicated the proof of this truth in the Proceedings of the meeting 
of September 27, 1919. 
