77J 



m+l — /J. I 



Vsy r .-^=^ . . . (6) 



r(m+l- fj) J 



0<7?(f,)<m+ 1. 

 These expressions (5) and (6), wliifli in a sense maj be considered 

 as each other's reverses, enai)ie us to arrive at (1) and (2). 



For this purpose we multiply l)0th members of (5) by a,„[ — ) r(l-P.) 



and afterwards summate from //(=.'>■ to wn = x (in the second memlter 

 under the integral sign). Let us write : 



r(i-;.) z'sy'" 



ƒ(,.)= ^- (-ira,,,—-^—^ - (,.-«)-+—, . . (7) 



05 /^izvw — a\" . , 



u {w) = :S a,„ /,„ (/sl/.r— a), .... (8) 



then we get already : 



T 



ƒ (.r) = r(i ~x) (^|-Jj (.«-£)" i" ƒ_, (^i/.:^) „ (Ï) dl R (;.) < i. (i) 



f iz \"' 

 If we multiply both members of (6) by a"' — , the first member 



by summation from m =z s to vi = oo passes on account of ^8) into 

 u{x), and if we execute the summation in the second member nnder 

 the integral sign, we find : 



4^0 = r^Y y(.«-s") ■' /_(i_,.)(LV*— §).9(5)d5, 



in which^(§)= È — — — ^~\ (g-a)'"-/' 



m=s r(W-|-l— fi) 



(f») 



<[ /? (fi) <[ *• -j- 1 (.•; positive integer or zero). 

 It clearly appears now that only on account of the special selection 

 H^zzX this expression on account of (7) passes into : 



1— i 



but we see at the same time arise as a condition /?(/'.)> 0; and as 



