32 DR GUSTAVE PLARR ON THE 



is decreasing, and so on. But as we assume 



and have generally 



and a&J"(a-k-t lt &c.)=f(0, &c.) = +1, because it reduces itself to its first term, which is 

 unity, we have finally 



a-) /£$-«*=§£=*• 



This formula (I.), which has been founded on the hypothesis of a = a negative integer, 

 is contained in Gauss's result, 



n(7-j8-q-l)II(7-l ) 

 1%-a- 1)11(7-/3-1)' 



in which the function Yl(x) is a transcendent of the same kind as the function T(l +x), 

 and in which the variable x represents any number, not necessarily integer. In fact the 

 case treated by Gauss relates to an infinite series in which neither a nor /3 are supposed 

 integers. 



We apply (I.) for the transformation of 



Writing 



V 7, e / ' 



J \ c J ~ L^<) J l«*/+l c te/+l gt/+l 



w T e assimilate the third member with 



811+1 in V( a '&*\ 



This gives 



8 = c — b, e = c; 



hence 



b = e—S, c = e. 



Thus we get the double sum 2 : 



V 7, e / L o ^o J i</+y/+ii»/+i e i»/+i 

 The interversion of the indices Z, w gives 



2 *£ 2'w = 2 "w2 "i. 



o u> 



We put 



t = w+t' , 



