447 



whence by repetition, and noting (3) 



k! 



f(r-l,x,k,0) = 



x(x + l)(x + 2)....(x + k) 

 and 



( — r) n k 1 



f(—l,x,k,n) = -. , ; ' -. - n = 0, 1, 2, 3 . . . . k—1 



J ' ' x(x + l) (x + k) ' ' ' 



therefore, since by (10), /( — l,x,k,k) = — xf( — \,x,k,k — 1), 



(12) f(—l,x,k,n) = ;. ; , — — n = 0, 1, 2, 3 , . k, but not n>k. 



x{x + l) {x + k) 



Example: 



x{x+l) (a+2) (x+3) (i+4) 2 (—1) 



2 (-D* 



8=0 



(41 i n 

 UJ x + i 



24 



when 



n = 





= 



—24a; 





n = 1 





= 



24a; 2 





n = 2 





= 



—24a; 3 





n = 3 





= 



24a; 4 





n = 4 



but = 240a; 4 + 840a: 3 + 1200a- 2 + 576a;, n = 5 



To find the value of /( — l,x,k,n) for n > &, set m = 1 in (9) and multiply 



through by 



k 



(x+l)(x+2) .... (x+k)/S(k,k) = 2 B{i,k)x°~^/S(k,k) 



i=0 



and set 



k 

 g(—l,x,k,n) for f{—l,x,k,7i)^B{i,k)x k ~ H /S{k,k): 



i=0 

 k 



g( — l,a;,&,7i+l) = A(k,n) 2, B(i,k)x ~~ l — xg( — l,x,k,n) 



2 = 



k,n = 0, 1, 2, 



Setting n = k, k-\-l, we verify that 



n k 



(13) gr(— l,aj,A,*+n) = ^ (— l) j_1 A(ft,A+n— j) 2 B(i,jfc) x* +i ~ i_1 



;'=1 i=j 



holds for n = 1, n = 2; and a complete induction shows, on taking account 

 of (14) §3, (p = n), that it holds for all positive integral values of n. On 



•See Chrystal: Algebra II, Ex. 26, p. 20. 



