450 



The corresponding results for n = k + 2, k + 3, etc., may be found by 

 putting these values successively for n in 



(21) /(— 2,x,k,n+2) = S(k,n) — 2x/(— 2,x,k,n+l) — x 2 /(— 2,x,k,n) 



which results from setting m = 2 in (9). The general result may be put 



into the form 



2/fc— 2 k— 1 



Z x 2k ~ J X D(i,j,k) S(k—i,n) 



(22) f(—2,x,k,n) = J ^— —\ k,n = 1, 2, 3 . . . 



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



in which the coefficients D, are independent of n: 



D(i,0,k) = 1 when i = 



= t = 1, 2, 3 



) 

 D(0,j,k) = 2 B(t, k—l) B(j—t, k—1) j = 1, 2, 3 . . . 



but I have not been able to determine a general formula for D(i,j,k) by means 

 of which to calculate the coefficients of /( — 2,x,k,p), p>k-\-l, without first 

 calculating successively those for n = k+2, fc+3, p — 1. 



By making use of (10) § 2, (21) may be reduced to 



2k— 2 A;— 1 



2 x 2k ~ j 2 E(i,j,k) S{k,n+i) 



(23) f(—2,x,k,n) = ^— -^^— — ; fc, n = 1, 2, 3 . . . 



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



with which compare (16) 

 Example: 



4 

 x 2 (x+l) 2 (x+2) 2 (x+3) 2 (x+4) 2 2 (-1) 1 ' (f) 7 -£— = S(4,n) x 8 + 



[12S(4,n) + SS(3,n)]x? + 

 [58 S(4,n) + 76 £(3,n) + 36 S(2,n)] x 5 + 

 [144 £(4,n) + 272 S{d,n) + 288 S(2,n) + 96 S(l,n)] x 5 + 

 [193 S(4,n) + 460 S(3,n) + 780 S(2,n) + 720 £(l,n)] x* + 

 [132 S(4,n) + 368 S(3,n) + 840 S(2,n) + 1680 S(l,n)] x 3 + 

 [36 S(4,n) + H2 £(3,n) + 312 £(2,w) + 1200 £(l,n)] x 2 

 n = 1, 2, 3 



