381 
1909-10.] Borel’s Integral and ^-Series. 
knowledge of the form of such a limit as 
Unlit J u „ + u l x+ . . + u n ~ !• , 
( n ! J 
in which the coefficients of of are numbers ^ containing ^-factorials 
(1 — q n ) ! etc., mingled with factorials of the ordinary type n(n — 1) . .(n — r-f 1). 
Although, generally speaking, we maybe ignorant of the form of such a 
limit as described above, we may have perfectly definite knowledge of the 
nature of such a limit as 
limit i 
U Q + Ul X + U 2^y { + 
+ u 7 
M 
ill 
in which [n] denotes the g-number (q n — l)/(q— 1) and the coefficients 
u 0 , ... . are g -factorials. 
Wherever in Borel’s theory the exponential e x appears, I propose, in the 
case of g-series, to substitute one or other of the following two series which 
reduce to e x in case q — 1. 
~E q (x) = limit J 1 + x + 
l+q 
+ 
. + 
M 
,} 
K<„) = l;». t {i + « t 'g 3 + g s . + .. 
HP 
Both series are continuous for all values of x from 0 to oo . The conditions 
for convergence are easily established. Wherever in Borel’s formulae the 
sign J of integration appears, we shall use §, the sign of g-finite integration, 
which is the operation reversing the g-finite difference operation 
AAfr) - *<«*>-*<*> 
qx — x 
= <f>'{x) . 
The successive differences are denoted (p"(x ), <p"\x ), .... In case the 
functions operated upon are differentiable, it is easy to see that in the limit 
q — 1, these differences become the successive differential coefficients, and g 
becomes J. With this explanation of the notation we write 
where 
= §E( - qx)u{x)d{qx) , 
o o 
u(x ) = U 0 + U X X + W 2jYj-j + 
E(-gcc) = l -qx + q s ^ - 
and 
