( 302 ) 
Mathematics. — ‘“Boren’s formulae for divergent series”. By 
Prof. J. C. KLUYVER. 
In his memoir on divergent series (Ann. se. de l’Kcole norm., 
t. 16, p. 77, foot-note) BoreL suggested that in his “adjoint 
integer function” 
perhaps the factor a,:2/ might be advantageously replaced by 
n 
p n hen 
Be +1), where p is a positive integer. 
In the present communication the truth of this remark will be 
shown. It will appear that this slight alteration in £ (az) leads to 
a “region of summability’’, identical to those found by Boren him- 
self, and also by SERVANT (Ann. de Toulouse, 2e série, t. 1, p. 152), 
when they considered other modifications of the adjoint integer 
function. 
Starting with the function f(z) and its expansion 
D D 
aj (2) == Er = Ms 
0 0 
which expansion we assume to have a finite radius of convergence, 
we consider the adjoint integer function 
P 
3 a Cn 2" a 
E, (ey 5 : 
0 
r (— + 1) 
Pp 
The integer p is arbitrary; if p be taken equal to unity, the 
function /, (a/z) becomes the function EL (az) of BOREL. 
In the first place it will be necessary to express E, (a/z) by 
means of a definite integral. A suitable path of integration W is 
obtained in the following manner. In the complex z-plane we draw 
a curve nearly in the shape of a cardioid, the cusp at z= 0 
pointing in the direction of «= — oo, and the curve itself enclosing 
the origin. We suppose that the path W begins at «= 0 and that, 
along the cardioid, it goes in a positive direction round the origin, 
ending again at «= 0, 
