8 
[102 
102] 
and also 
Multiply! 
Cn 
102. 
ON A DOUBLE INFINITE SERIES. 
[From the Cambridge and Dublin Mathematical Journal, vol. vi. (1851), pp. 45—47.] 
The following completely paradoxical investigation of the properties of the function 
r (which I have been in possession of for some years) may perhaps be found interesting 
from its connexion with the theories of expansion and divergent series. 
Let S,.<jbr denote the sum of the values of <£>r for all integer values of r from 
— 30 to qo . Then writing 
u — [n — x n ~ 1 ~ r , (1) 
(where n is any number whatever), we have immediately 
^ = 2 r [n — l] r+1 x n ~ 2 ~ r = £ r [?i — l] r x n ~ l ~ r = u ; 
that is, ~ = u, or u = C n ef, 
dx 
(the constant of integration being of course in general a function of n). Hence 
C n & = 2 r [n — l] r x n ~ l ~ r ; (2) 
or e* is expanded in general in a doubly infinite necessarily divergent series of fractional 
powers of x, (which resolves itself however in the case of n a positive or negative 
integer, into the ordinary singly infinite series, the value of C n in this case being 
immediately seen to be Tw). 
The equation (2) in its general form is to be considered as a definition of the 
function C n . We deduce from it 
X r [n - l] r {ax) n ~ l ~ r = C n e ax , 
[nf - 1 ] r ' (ax') n ~ 1 ~ r ' = C n >e ax '; 
(where r, 
p being 
multinom 
in the eq 
In j 
and if p ■■ 
or puttin 
Now 
(at least 
then 
and 
1 
Om+n- 
1 
.¿m+n- 
by 
or 
C.