80 Mr Kienast, Proof of the equivalence of 



VIII. The relation connecting Cesaro's and Holder's means 

 can be deduced in the same way. We have 





L\ = i 



n(n + l) n{n+l) 



= 2hf — -1—lxhf. 



« w(w + 1)a=i " 

 Assuming therefore 



o(«) I n 

 ^ =C„ h'^:^ ^ r^ td, h['^ (8), 



we fand \ "[ K J ri+i,^K ~^^j^^^.«+i^ • 



Hence c„, „+i =(« + !) Cn+i, « , 



or Cn,K= k\. 



Starting from the numbers (4), we have 



sr'=... = sr_\=o. «r=i. -. *1"'=(""^+''). ■••; 



?"=... = si 



as is easily verified by the formula 



.ti ^ ™,roV « / V a: + i ;■ 



Writing (8) for 7r = X, we find 



d),^,= \(\+l) ... (X + «-l)-X«^0. 



Starting with the numbers (6), that is with the numbers (4) for 

 X,= 1, we have 



and from formula (8) follows 



A = l V K 



