383 
1909-10.] BoreFs Integral and ^-Series. 
and provided we have knowledge of the limit 
K X ) = L \u 0 + U 1 X+ . . + U n Ar. f 
n oo ( pij I / 
we write 
u n = §E( - qx)u{x)d{qx) 
0 0 
For illustration consider the following examples : 
l-[2]f+[3]f 2 - .... 
Here 
u(x) = 1 - [2]to + - . . . . 
= (1 - qxt)1& q ( - tx ) 
and the ^-integral (6) gives us 
oo oo 
^E( - qx) E g ( - tx) - gj^E( _ Q. x )^q{ ~ bx)tx.d(qx) 
o o 
1 qt 
l+t (l+t)(l+qt) 
1 
(1 + £)(1 + qt) 
• ( 6 ) 
^<1 , t < 1 
which is correct, as may be easily verified in other ways. If t = 1 q = 1 the 
value is agreeing with Borel’s sum of 1 — 2 + 3 — . . . . 
Example (B) 
l-[2] 2 * + [3] 2 * 2 - .... 
gives an integral 
§E( - qx) E 2 ( -xt){ 1 - (< q 2 + 2 q)tx + qH 2 x 2 }d(qx) 
_ 1 (g 2 + 2q)t [2]qH 2 
t + 1 ( 1 + £)(1 + (^) (1 + £)(1 + <2^)(1 + q^t) 
_ 1 - qt 
(l+t){l+qt)(l+q 2 t) 
which is easily verified. In case q = l,t = l we have BoreFs sum of 
1 — 2 2 + 3 2 — ... -0. 
IY. 
Asymptotic formula — 
In the case of an asymptotic series 
a 0 + ajx + ajx 2 + .... 
we suppose that f(v) is an associated series 
f(v) = a 0 + a 1 v + a 2 -^ ] + . . . . 
