449 



In (8) set t = —2, k = 1 



x/(— 2,x,l,0) =/(— 1,3,1,0) +/(— 2,z+l,0,0) 

 whence by '(12) and (3) 



/(— 2,x,l,0) = * + ^-^"2 



.r-(.r + l) x(.r + l) 2 



1! y 



•r-U+ l) 2 

 Again, setting & = 2 in (8) 



l 



(1+0 5(1—1,-1) x* 



f(— 2,.r,2,0) = -/(— 2,a;,l,0) + --/(— 2,x+l,l,0) 



.r x 



= — ttt? 5T5 2 (1+i) 5(2—1,2) X 1 ' 



x 2 (x + l) 2 (.r + 2)- f=0 • 



Assume 



(18) f(-2,x,k,0) = ,. ... A , .., 1 (1+i) B(k-i,fc) x ! 



x-(.r + l)- (x + k)- - =0 



and a complete induction, on taking account of (11) §3, shows that this holds 

 for all positive integral values of k. 



Therefore: 



*/ i 



Co = ^77 ,,, : r-^ 2 (1 + i) B(fr— l,k) I* 



.r 2 (x + l) 2 U + £) 2 - =0 • 



— ci = -^ — ttt-^ 7 — r^ 2 B{k—i,k) x l 



* 2 v* + l) 2 (* + *) 2 i=0 



and 



(19) f(-2,x,k,n) = ^ A~ X) " *"' , lM , S (1+i-n) B(*-*,t) x 1 ' 



x-(x + l) 2 (x+k) 2 - =0 



A- = 0, 1, 2.. .; n = 1, 2, 3 k— 1 



On computing, by means of (10), the values of /( — 2, x, k, k) and 



/(— 2, x, k ,k+ 1), we verify that (19) holds for n = 1, 2, 3 fc+1 



but not for ra>ft+l, 



Therefore, 



^om 5" / i\i W * M {-x)"k! V /i i • \ n>/; -n * 



(20) 2, ( — 1) •, — = — +- — '- — - Z (1+t — n) B(k — i,k)x 



i=0 UJ (x+i)> x 2 (x + l) 2 (x + k)- i=0 K 



k = 0, 1, 2, ; n = 0, 1, 2, fc+1; not «>/.-+l 



29— 49GG 



