ON THE THEORY OF NUMBERS. 359 



The formulae in this Table relating to f a (x, x), f(x, x), and f(x, -x) 

 require a certain modification, when n is a perfect square*. If, on this hypo- 

 thesis, we represent by f,(.r, x) the function obtained by writing x=v 8 =u li 



in '^'l'J > f *(*> •*).** of the order *(*)■+*(«)— h and is divisible by 

 (^-.i-)*W-*(»)-i. Again, if »=„' and (-)=!, we represent hm ■#"' *> 

 \u=v=x] by f (x, x) ; this function is of the order *(») + ¥(n)— 1, and is 

 divisible by ,v*(»)+*(»)-i x (a; s — l)K*(«)-*(n)+*'(»)-*'(»)]-i . jf (^L\ = _\ we 



represent lim ^"^ [»=v=afj by t(x, -a?), and this function is of the 

 order *(«)+¥(«)— 1, and is divisible by 



iB »(n)+* (B )_l x (V — l)H*(«)-*(n)+*'(H)-*'(»)]-l - 



The formula IV. may be deduced from the equation 



/.(*» *)*/■(** l-*)xa*<»)//*,l)x(l-*-)*<">/ 8 C*, -r^-) = 0, 



pf which the order (after division by powers of x and #— 1) is shown bv the 

 Tabletobe2*(n) + 6¥(n). 



In proving the formula V., we might have employed the equation 



tf* (n) / 8 U, -j=0 instead of / s (>, 1 -#) = (). If, instead of the former equa- 

 tion, we employ the two a/*WfJar,±\=0, x*WfJx, --\ = 0, we obtain 



the formulae Y. and VI. simultaneously. 



Lastly, the formula VIII. depends on the equations f(x, x) = and 

 f(x, —x) = 0. 



134. Connexion of the formulae of M. Kroneeker with Elliptic series. Re- 

 searches of If. Hermite.—&. Kroneeker has given a remarkable analytical 

 expression of the formulae IX. and X. (art. 130). He employs the identical 

 equations 



00 00 n 



2[2 + (-l)»]X(n) 2 »=2 2 , 



l K n i[i+(-i)Y?' 



GO ±Ltl OC „» 2 +»-l 



|S¥(4h+1) 2 * = 2 iS_i 



a»-i\a' 



i ia-2 2 "- 1 ) 



oo 



(l + 2q + 2q i + 2q 9 + . . .) x 2.E(«) 2 » 





 oo 

 = 2[E(>i) + 2E(n-l 2 ) + 2E(n-2 2 ) + . . .] 2 », 



.(%*+2g*+2 2 ¥+...)xS , (»)2" 



1 



.00 



=2 2 *S[F(»)+F(»-l,2)+F(n-2.3)4-. . .] 2 ", 



of which the first two are immediately verified by expanding their right-hand 



* The necessity for a corresponding modification of the formula; IV. and VII. is obvi- 

 ated by the assumption G(0)= — -j^. 



