461 



u=h~l 



^^ 2n/ '^' ' Zb;(2« + 2A)/ 2„ '^' 



where v now stands for — . 



Bj means of this equation ?(2/< -{- \) is expressed in terms of 

 C(3), ?(5), . . ., ?(2/< — 1) and taking successively h = 1,2,3 .. ., we get 

 S(3, ?(5), 5(7), . . . expressed by a linear combination of expansions 

 the index of which is yV ^)- 



A slight transformation of (6) is possible. By using 



^^''~ 2^^ 2^1 '2^' 



u = 1 



and by effecting some other reductions, it may be shown that (6) 

 can take the form 



n=A — 1 



{h -«) (2/i 4- 2n — 1) (— l)"-i v^" 



^ h(2h— l\ 2n! ='^ ^ ^ T- 



n=2 



h{2h—\) 2n! 



2h{2h—\)} 2/i/ ^4-/(2n + 2A)/ ^ 



*"^ 3 



If we put A = J and /* = 2, it will be seen that 



g(3) = 3;— t,'_y" .^^ü (2n + 3)y2n-|-2[ 



/ 4 / .*— ' (2n 4- 4) / \ 



I will now proceed to show that for each of the quantities C(2/i -f- 1) 

 there exists a linear combination of expansions with an index less 

 than -^. For this purpose I use again the identity (4) and writing 



w=oo 



15 ^i— ' n« 



n = l 



^) Similar results were deduced by Mr. van der Gorput in the paper quoted. 

 However, in the fundamental expansion of the quantity I{n,a) on p. 1464 by a 

 slight inadvertence the factor 22k Jias been omitted in the general term, hence 

 in all the subsequent expansions the general term should be multiplied by 2^^ 

 and the index of the series on p. 1470 is -gig- and not y^f. 



