267 

 The last factor is not equal to zero, for then the sum 



^ Xv (^, f^ e ''^ = {i{D„) ifiD,,) Xv (7,, --]2: Xv {m, k) e" 



1=1 V ^njm=i 



m=l 



would be equal to zei'O if /i = 1, 2, 3, . . . , ^^ that is the equation 



k _ 

 ^ Xv (^1 ^) ^"' = Ö 



2 TTJ» 



would possess the root zero and k different roots e ^ , and this is 

 impossible, since the coefficients Xv 0>'> /^') ^^'^ "Ot identically equal 

 to zero, according to 4, HI. 

 Consequently we conclude 



i .4, ft (i>„4.;V/ (!>.+ ;.) Xv (« + ;.,-^^) = 0. 



;=o V " + •^y 



Proposition 8. If k be squareless, then we have 



;,— 

 Z)'„4-; being the G. C. D. of n -\- I and k. In this formula n may 

 be any integer number whatever. 



(This formula is to be found in the above mentioned article of 

 Prof. J. C. Kluyver). 



Proof. Make in proposition 7 i' = 1 ; then we shall have 



and 



k 

 according to 4, IV, since n -\- I and — have no common factor, 



because k is squareless. 

 Hence it follows 



Proposition 9. If k be an odd squareless number, then we have 



I J being the symbol of Legend re. In this formula n may be 



any integer whatever. (This foi-mula is also to be found in the article 

 of Prof. J. C. Kluyver). 



