Mathematics. — “On the necessary and sufficient conditions for 
the erpansion of a function in a Binomial Series”. By Dr. 
H. B. A. BockwinkeL. (Communicated by Prof. H. A. Lorentz). 
(Communicated in the meeting of May 3, 1919) 
PincHERLE has given a necessary and sufficient condition for the 
expansion of a function in a binomial series (Binomialkoeffizienten- 
reihe)*). It runs thus: 
The necessary and sufficient condition that an analytic function 
w(a) may be expanded in a series of the form 
ae at 
wo (@) = Sn 0 (7) ) Sed oe ae 
0 
is that w(x) be coefficient-function (fonction coefficiente) of 
another analytic funtion q(t), which ts regular and zero at infinity 
and whose singularities lie all within the circle (1,1), with centret=1 
and radius r=1, or on the circumference of it, provided that, in 
the latter case, the order of g(t) on the circumference, taken in the 
sense defined by Havamarn, be finite or negative infinite *). 
By a coefficientfunction w(z) of an analytic function g(t) of the 
kind mentioned PincHERLE means a function which can be deduced 
from g(t) in a more or less simple manner, according to the order 
of g(t). The relation between the two functions is, however, always 
such that conversely q(t), called by PincuerLe the generating function 
(fonction génératrice) of w(x), follows from w(x) by the equation 
a + 1 
po ye. op nest an ema 
ae 
This means: the coefficients of the series of negative integral 
powers of ¢, in which g(t) may be expanded in a neighbourhood 
of to, are equal to the values of w(x) for positive integral values 
of «; the name coefficientfunction for w(x) is due to this circum- 
stance. 
The question now arises, how it must be discriminated if a given 
function may oe expanded in a binomial series. This question is not 
IS. PINCHERLE, “Sur les fonctions déterminantes’’, Annal. de Ecole Normale, 1905. 
2) A circle with centre ~ and radius 7 will be denoted by (a, 7). 
