76 Mr Kienast, Proof of the equivalence of 



Proceeding to build up the expression for s[l'^^\ applying the 

 same transformations as above, we find 



"kAC^ - ") K'-"- (X - « - 1) Att'-M 



n-1 



„(«+!)_ 1 'V 



o(«+l)_. ^-'^-lz.('c+l) I'^v'^ 7,(«4-l) 



from which we conclude 



K't'l^-^ ^ d.,.+A''-'' (2); 



n K i- /ON 



(^n, K4-1 ^= Cn, K V"/) 



and a series of relations involving the numbers c?a,k and cIk^k+i- 



Equation (2) is of the same formation as (1), and therefore (1) 

 gives the required expression of s^^''^ by the numbers h'-j^K 



II. Since Cn i = -^^ , (3) leads to 



n 



Cn, K = n~'' (n — K)(n-K+1) ... (n — 1), 



from which follows, for k = 2, Cn^= ~ ~ , which is in 



accordance with the expression for s^^^ above. 



III. c?A.,K may be determined in the following way. Putting I 



ai = a2= ...=aA-i = 0, a^ = l, aA+i=... = (4), 



we find 



s(0)= =,.(0) _A (0)_-, (0) _-, 



^(1>- - o(l)_A Jl) _ 1 „(1) _ 



^K ^' 6a4-1 — > , T> *A-4-S! 



+i~\+l' ^+2 X + 2' 



o(2). 





^+1 ' ^+2 (\ + l)(X + 2)' 





^+'' (\ + l)(X + 2)...(\ + /c)' 



