Mathematics. — “On a remarkable functional relation in the 
theory of coefficientfunctions’. By Dr. H. B. A. Bock winker. 
(Communicated by Prof. H. A. Lorentz). 
(Communicated in the meeting of September 27, 1919). 
1. Let g(t) be a function having no singular points without the 
circle (1,1), i.e. the circle with centre t=1, and radius 1. Let 
g(a) be zero and the order g of p(t) on the circumference of the 
circle (1,1) be different from —+ oo. Then in the series 
Ssh, De 
gp (t) == Dn fl ; 5 e e A @ ° 4 (1) 
0 
the characteristic k= lim [log g, :logn] of the coefficients g, is 
n= 
also different from +o, in virtue of the known relation k—=g—l. 
If g <0, then the integral 
vO fy (ie sige olene zakt a (2) 
a JEL 
(1,1) 
taken along the circumference of the circle (1,1) evists for R(x) > 0, 
because in that case the series (1) converges along that circumference: 
the value of #1 in it is so defined that the argument of ¢ lies 
7 
zie 
with zero limit. The function (x) is called coefficientfunction by 
PINCHERLE'), Owing to the relation g, = w(n+1). Conversely p(t) is 
called the generating function of w(x). PixcHertr considers the 
relation between these functions especially from the point of view 
of the functional calculus. If we write 
lope Slower vera artist 
Lis an additive functional operation, which satisfies a certain number 
of simple functional relations; these relations may be used in order 
to define the coefficient-function in those cases in which the integral 
(2) does not exist. Thus we find easily 
Iltg()]=o(e¢+ 1), and 1 p'(l) = —(#—1) w (el). . . (4) 
and by combining these two equations 
1[¢'()] = — ad) Lt 0), 
1) Sur les Fonctions Déterminantes, Ann. de |’Ke. Norm. (22) 1905, (Ch. IV). 
of 
continually between — 5 + d and - J, d being a positive quantity 
