382 
Proceedings of the Eoyal Society of Edinburgh. [Sess. 
For the case of an asymptotic q -series 
a 0 + ape -1 + a 2 x~ 2 + . . . . , (a?>0) 
we define the sum as 
I use the factor d(qt) merely to denote with respect to what variable (t) we 
are operating. 
The reader may easily verify the following : I omit constants, also d(qx) 
under the finite-integral sign, since all integrations are with respect to x. 
Ax n = [w]af -1 , = # n /[w] 
A(aj + l)(x + q).(x + q n ~ l ) = [n](x + l)(a + q).(x + q n ~ 2 ) 
AE(aaj) = aE(aqx) 
S iE(aga) = ^E(a«) 
III. 
SE(-g*)= -E( -x) 
E g (g)E(aa;) = 1 + (1 + a)x + ( 1 + 1 + ga) ^ + . . . . 
l z i ! 
E 2 (a)E(a#) = ! + (! + a)x + 
all of which will be required in the following work 
§z n E( - qx) = [tt]j§aj n-r E( _ q X ) = \ri\ ! . 
o o 
• ( 5 ) 
for the first term on the right-hand side of (5) vanishes, both when x = 0 
and when x= oo . We see that 
^ ”M ~ 
