83^ 



27rf , , 2-0!+; K / 27r(n-0 



.=,1 V y V , . ^ ^j^^ 



2-x. 



_ r A j ^(1) + 2 'Vc-0.5 'Innlfio. ■/ j , (a,=:4/r— 1) 



r r «=i V /) 



If ill both members of these equations the functions /'are expanded 

 into series, the summations indicated are to be executed still further. 

 1 shall, however, [)erform these reductions onlj for special values 

 of the parameter ^, in consequence of which the general results are 

 simplified. 



In the equation (III) I substitute therefore S = i, at the same 



time I replace ^ by -- and accordingly y by 2y. I further write 



e /^ = 9, e 'i =. q'. 

 The numbers q and q' are then positive and smaller than 1; 

 they satisfy the relation 



jt' 



lop qXiogq =— , 

 a 



but are for the rest arbitrary. 



In this way the equation (III) passes into 



"^"(-1)" ƒ('?-"+') = ' -^"^"(—1)" /('?''"+'), 

 )i=o y '(=0 



and if the functions ƒ are expanded into series, we shall tind 



■^ g m=i V«;i4-?-'" 9 >n=i \ajl+q'-^»^ 



In the equation (IV) I substitute 5 = 0. We have then in the 

 first place 



/(I) + •/l/C'z^'O = '^- !/(i) + 2'5/(7'2") j, 



71=1 r ( «—1 ) 



and if again use is made of the expansion into series of the functions 

 ƒ we find 



q { >„=i \a J 1—92»* 



=:hoa~^^\,m + 2 2 (-)^-^i {a = iw~l). (VI) 



The equations (V) and ^VI) completely symmetrical with regard 

 to 7 and q' are again conspicuous for the remarkable properties of 



