364 Generating Functions in the Theory o/ Numbers. [Nov. 23, 



The product 



Xi&X,*. ....X> 



offcen appears in arithmetic as a redundant form of generating func- 

 tion. The theorem above supplies a condensed or exact form of 

 generating function. 



Ex. gr. It is clear that the number of permutations of the ^ 

 symbols in the product 



which are such that every symbol is displaced is obviously the 

 coefficient of 



#1^1 #/3 .... #,, 



in the product 



(x 2 + . ... + x n )h(xi + x 3 + .... + xrf* .... (aJi + o^-f- ---- + #n_i) n , 

 and thence we easily pass to the true generating function 



_ 1 __ 



3*2x1X2X3X4. .... (nl) x l x t ---- x n ' 



In the paper many examples are given. 



Frequently the redundant and condensed generating functions are 

 differently interpre table ; we then obtain an arithmetical correspond- 

 ence, two cases of which presented themselves in the " Memoir on 

 the Compositions of Numbers." 



A more important method of obtaining- arithmetical correspon- 

 dences is developed in the researches which follow the statement and 

 proof .of the theorem. 



The general form of V M is such that the equation 



gives each quantity x, as a homographic function of the remaining 

 n 1 quantities, and it is interesting to enquire whether, assuming 

 the coefficients of Y arbitrarily, it is possible to pass to a correspond- 

 ing redundant generating function. 



I find that the coefficients of V must satisfy 



2 W 'n? + n 2 



conditions, and, assuming the satisfaction of these conditions, a re- 

 dundant form can be constructed which involves 



n 1 



undetermined quantities. In fact, when a redundant form exists at 

 all, it is necessarily of a (n l)tuply infinite character. 



