378 Proceedings of the Royal Society of Edinburgh. [Sess. 
XXII. — Borel’s Integral and g-Series. By Rev. F. H. Jackson, M.A. 
(MS. received November 29, 1909. Kead December 20, 1909.) 
I. 
In his Theory of Infinite Series , Dr Bromwich gives an account of the 
recently developed theory of non-convergent and asymptotic series, so far 
as the arithmetical side of the theory is concerned. The connection between 
Borel’s integral “ sum ” and Euler’s well-known transformation 
^ju n x n =u y y + (A^)jy 2 + (AX)?/ 3 + ; • • . (1) 
y = x/( 1 - x) 
is discussed. Now, if we apply this transformation to such series, for 
example, as 
1 - b 1 - qb 1 - fib 
+ 
x bx 
1 - x 1 - qx 
+ . 
(?“- l) x+ (<] a ~ 1)(2“ 
(3-1) (3-l)(3 2 -l) 
^ \<: L H- . 
( 2 ) 
(3) 
which are of great interest in the theory of Elliptic Functions, we obtain 
results which may be described as formless, or at least of such complexity, 
owing to the mixture of ^-factorials (1 — q n ) ! with ordinary factorials n\, 
that the resulting series are practically useless so far as the possibility of 
applying further transformations is concerned. In fact, when we are 
dealing with power series, in which the coefficients are g-numbers, we must 
use a modification of Euler’s transformation if the results are to possess 
that quality of form which will make them interesting and useful. The 
required modification of Euler’s transformation is 
U'X" m 
U, + 
Atq + ■ 
(I-*) 1 (1 - x){\ - qx) 1 (1 - tf)(l - qx)(l - (fix) 
A (2, iq + . . 
in which 
A\ = (D-l)(D-(i) • • (D . . . (4) 
D u n = u n+x , 
reducing to Euler’s series incase q = 1. If we apply this transformation 
to the above series (2) and (3) we obtain from (2) the well-formed series 
* bx 2 (g - 1) 
(1 - x){\ - b ) (1 - a;)(l - qx).( 1 - b)( 1 - qb) 
+ . . 
