Mr Mathews, Reduction of Generating Functions, etc. 69 



Reduction of Generating Functions- by means of Complex In- 

 tegration. By G. B. Mathews, M.A., F.R.S., St John's College, 

 Cambridge. 



[Bead 13 February 1905.] 



In solving various Diopbantine problems we have to perform 

 an operation 12, which may be defined as follows. A rational 

 function' of a and other variables is formally expanded in positive 

 and negative powers of a ; the negative powers of a are then 

 discarded, and a is replaced by unity in the remaining terms. 

 If <j> is the function in question, the final result is indicated by 

 O a <£, or simply by ft</>, when there is no risk of ambiguity. 



Thus, to take a very simple case, let 

 i 



(1 -■«-)(!-*) 



= (1 + ax + a 2 x 2 + . . .) f 1 + - + — n + . . . 

 \ a a- 



= taP-VxPyi : 



then it is clear that 



Xl(/> = 2«*y9 



with the condition that p ^ q. By putting p = q + r, we obtain 



n<p = 2 (xu)Vx r (q >0,r> 0) 



1 

 -{l-xy){\-xy 



but in more complicated cases it is not so easy to find the reduced 

 value of 0<£. The object of this note is to show how the principles 

 of complex integration may be applied to the problem. 



Suppose that, | a \ being greater than 1, we have a convergent 

 expansion 



<f> (a)= ... + b,a~" + 6 1 a~ 1 + c + c 1 a + c 2 a 2 + ...; 



