134 



Dr Kievast, E.vtensioiis of 



3. I proceed to prove some other identities. We have 



,(1) 



i KQ 



K 

 1 



a« = - r -/ 



U<i>_-,.(i' ! 







,(^) 



,(A-|-1) 



1 K " 



M) 



Jl + 1 M 



.(7); 



and by successive substitution we find 



1 



n ' " 



(3) Q,„.,(3) , O/., 1\^(3) /„, 0\ ,,(3) 



" ' - n—l> n + l n 



+ /2) +1|,.C2)_,,(2) } 



= (n + 1) C - 3m'- , + 3 (.^ - 1) r^' - {n - 2) ,-- ^ 



fl '■ n n-l> 



Writing 



, . (n + X - yu, - 2) r 



(\+k) ^ ^ 



M+A — M — 1 K.K.m 



.(8). 



we can easily verify that 



6r, =tT>.i — (X — l)0. ,, 



\,K,n. \ + l,K,n ^ ■' A,K+l,w 



Moreover 



Developing a^ in this way we obtain, after a finite number p 

 of steps, the formula 



cin= S 



A+K = P 



A, K A, K, n \^ 



iUp)_, 



.(p) 



.(10). 



The upper index of all the 7''s is the same throughout this expres- 

 sion. For the present purpose it is not necessary to determine the 

 coefiicients c^ ^, which are integers. 



In consequence of the definitions of s and r we have 



7>) = (n>vi). 



