GENERATING- FUNCTIONS IN THE THEORY OF NUMBERS. 
151 
Art. 57. Of order 2 we Lave the product 
and in performing the multiplication we find a term involving 
and if be not a function of Xi and x.^ the terms involving x^'x^'^ can only arise in a 
manner similar to this. 
If, however, be such that olhX^ is a multiple of x-^, and consequently xja^^ ^ 
multij)le of we at once get an addition to the coefficient of x-^^^x/k In the present 
case the coefficient becomes 
Hence, considering monomial values of inequality 
must be satisfied in assigning to a function of x^ and x.^. 
We may put subject to the above condition, equal to any monomial integral or 
fractional function of x-^ and x^. 
We may 7 iot put 
wherein n differs from m. 
Art. 58. Of Order 3, the particular redundant product is 
+ + «a35^a3 
V ‘^ 12^23 
and we must realize the coefficient of 
