apply successfully the identity 
k=m—h+j 
Bn k= = Bj Bn—k, h—j + 
Here j is rather arbitrary, only it must be taken > 4—1. 
The function 1:1—e is uniform with the single pole «==1, so the 
regions Gy and G, are determined with the aid of the single equation 
ag (Hpi 
and inside G, the mark of the series is always indicated by 1: R,. 
We need not look about for possible infinities a of g(u). 
FW (0) 
a we obtain out of 
Ee 
By the introduction of the coefficients 
the series (V) a series for #(«) and it is apparent that we shall 
now be able to find the various parts 2; of the region of divergence 
of this new series by applying successively to the region of diver- 
gence of series (V) the transformations 
(a; A; z); 
In a similar way the regions Gj’ and G,' can be determined. 
However we must keep in mind that in accordance with the nature 
of the function F(x) the mark for the new series will perhaps be 
in some regions no longer 1: 2, but 1: | a |. The case might present 
itself that notwithstanding the existence of a region of convergence 
for the series (V), the deduced series for /(«) might diverge every- 
where, 
The preceding more general considerations on series of polynomials 
we do not desire to continue before we have treated some simple 
examples. It seems difficult to make a simple calculation of the 
coefficients Bm agree with a large extension of the region of con- 
vergence. But if on this last point we are not too exacting we can 
obtain rather useful developments which are suitable for application. 
Supposing that a is a given constant, we put 
pa ee (a—1) a4 
a—u 
The function g(u) is everywhere uniform, so in this case k=; 
inside the circle |u| = is the single pole u = a, 
