different mean values 77 



and /,;*»=... =/t(VO, ^^=1, h['i=l, ..., 



n^ ... ^_i ^, /i^ ^, z^+i ^^-^, ..., 



Writing equation (1) for n = X + K, and substituting these special 

 values, we obtain 



f \(\ + l)...(X + «-l) V^ ] 



rfx,.-(^ + «)| (\ + Kr (\+1){X+2)...(X + k)\' 



which is, for /c = 2, in accordance with the above expression for s^^\ 



IV. Lemma a. The coefficient c^a,k is a positive number for 

 X = l, 2, ...,/c = 2, 3, .... 



It is easy to verify the inequalities 



^4^^—^'— (y^ = 0,l,2,...,.-l), 



X+ K X + K — fl 



from which results, by multiplication of all the left-hand and all 

 the right-hand sides, 



X{X + 1)^. .(x± «-l) > '>^ 



(X + kY ' -{X + 1){X + 2)...(X + k)' 



which demonstrates the assertion. 



Lemma ^. The coefficient c?a,k satisfies the equation 



lini|<.-^(«-l)/c(« + l)i| = (5). 



To show this we expand c?a,k in the form 



and the proposition follows. 

 A consequence of (5) is 



v. 



lim ]S f^A, K = 00 , 



)l-*-x A = 1 



