ill combinatory analysis 213 



Hence, arranging (2) in terms of t^C'i, 776*2, ..., we obtain 

 V -V 



' m ' in—\ 

 \ — X 



= a.'"^-i {(1 - ^.n-«i+ig»i-"i+i) _ .^H^«+?n (1 _ ^^n-m-H ^jin-sm+i^ r]Ci+ ...} 



= a;'''-'vVn-n,+i (3). 



CO 



If we write v^^ TI (1 — *•(/'■)= F,,^ (4), 



>- = 



then (3) becomes v„, — v,n-i = ^''"^~'^ vVn-m+i (o). 



It should be observed that Fo and Vo vanish identically. 



In particular take n = 2,m = l, a,nd n = 2, ni = 2. We then 

 obtain Vi = 7]Vo, V2 — v^ = xrjv^ ; 



and so Vi — 7]Vi = ocqr)'-v^ (6). 



Now let Vi = l + a^x+a2cc-+ (7). 



Then from (5) 



1 + aiX + a-iO? + . . . — (1 + a^xq + a^xif + . . .) 



= a-^ (1 + tti^y/ + aojf-(f + ...) ; 



and so a^=—l~^ (^2 = p. ry^ ^ (8). 



1-q (l-q)(l-q') 



But when x = q, C,- = 1 ; and so 



V, = {l-q)-q^{l-q^) + f^{l-q^)- (!)). 



From (4), (6), (7), and (8) it follows that 



^l-q^(l-q)(l-q'^)^ 



^ a-q)-qH ^- q') + q''('^-q')--- .. ^^ 



(l-5)(l-2'^)(l-f/)... ^ '^- 



Similarly we have 



and, when x — q, 



and V, = {1- q') - q^ (1 - g«) + 5" (1 - f/«) - ... . 



\ ^ X X'Q^ 



77 1-^ (1- 7) (!-(/-) 



l-q (1-g) (!-(/) 



