•268 



THE REV. F. H. JACKSON 



The coefficient of x 2 '" is c m 2 , which is determined by the recurrence relation 





(l-g*») 



"2 a -i C m-2 T" —£ a 3 e m-3 +•••• + 



1 ) 



2^T 2 a n c m-n j ' 



. (35) 



J m = x m {c m+<) c + c m+1 cx l i + c m+2 c Q x i + }, . . . (36) 



in which the coefficient of x'" +2r is c m+2 ,..c. 2r , in which c lll+2r and c 2l . respectively are 

 determined by the above recurrence relation (35). 



§ 4. Expression for O n (x). 

 We have now established the form 



U \ 1 + 2q im x Pl cos 20 + q im x> Pl 2 i S j 1 + 2g 2m *p 2 cos 20 + <f n a; 2 p 2 2 1 



m=0 I J 7)1=0 I J 



n •! 1 + 22 2m zp n cos 20 + 2 4m x 2 Pr! 2 !• , = J + 2 J x . cos 26 + 2 J 2 . cos 40 + . 



m-0 ( J 



• (37) 



the constants p 1 p 2 . • • /o„, being connected with the constants a l9 a 2 . . . . a„. para- 

 meters of the J functions, by the relation (1 +/) I a;)(l + p 2 x) . . . ( 1 + p n x) = 1 + a x x 



+ a 2 x A + . . . +a n x n . 



Thus, if we choose p 1 = p 2 



1 , a 1 = n, a 



_n(n— 1) 



2! 



put g~ l for x, we shall obtain the n th power of Jacobi's © function 

 @ V2K0\ I n(l- 2 *»)iTj + 2J 1 cos25 + 2J 2 cos4e+ . . 



in which 



a n = 1 , and 



• (38) 



• (39) 



j _ , , ra 2 ra 2 {w 2 g' 2 + rag 2 + ra 2 - ra} 2 + c r 2 



° - <z 2 (<z 2 -i) 2 5 4 (? 4 -i)% 2 -i) 2 7 r • • ■ . , • 



the other coefficients being determined by means of the recurrence relation 



J»' = g-'»{c m c + ( ? - 2 c m+1 c 1 + g- 4 c m+2 c 2 + } (40) 



The simplest case of the product II^o^ . . . a n , x) is when a x = a(g — l)/q, and a 2 a 3 , 

 . . . . , are each zero. In this case the above product and series become 



which we saw in (9) was a solution of Ay = ay, and was there denoted E 5 (asc). This 

 product and series is, of course, well known as 



( 1+ *)H) 



1 + 



(2-1) (q*-l)(q-l) 



If in the series E g (ax) we replace g by g -1 , which we term inverting the base, we find 



ax . a'x 



2,,.2 



E q -i(az) = l + ™+q™+q 



s~z 



x a x 



[1]'*[2]! * [3]! 



