445 



§4. 



The properties of 



f(n,x,k) = 2 (-1) 1 ' [*] (x+i) n 



1=0 



(k) 1 



i = 



and an application of — , ( — l) 1 • — : in the theory oi gamma functions 



x + i 



suggests the generalization: 



(1) f(.t,x,k,n) = 2 (-1) 1 ' [J] i n {x+i) 1 



k,n = 0, 1, 2, 3 ; f = 0, ±1, ±2, 



Whence 



(2) f(0,x,k,n) = S(k,n) k,n = 0, 1, 2, 



(3) f(t,x,0,n) = x when n = 



= when n > 



(4) /(i,x,l,ra) = x — (x+1)' when n = 



= — (z+1) when?? > 



When i < 0, this fimction has poles at x = — 1, — 2, — k, and 



also when n + t < 0, at x = 0. 



Since f(t,x,k,n) = 2, (—1) | -j i (x+i) (ar+i) 



we have the recursion formula 



1-0 ^ 



(5) f(t,x,k,n) =2 • z/U — m,x,k,7n-\-n — i) 



i = ^ j 



f = o, ± 1, ± 2, . . . . ; k,n = 0, 1, 2, 3, . . . . ; m = 1, 2, 3, ... . 



If t is not negative, we have on setting t for m in (5) 

 ( . . 



(6) f(t,x,k,n) = X J- | a; 1 ' S(£,i+n— i) fc,n,i = 0, 1, 2, 3 . . . 



i=o lM 



If < n ^ k 



2 (— 1)* fj] i (n) (x+i) ( = (_i)V B) /(*,*+«,*-», 0) .n = 1, 2, 3 A 



i = ^ J 



Wh ence, making use of (2) §3, 



(7) f{t,x,k,n) = 2 (— if A(i,n) k U) f(t,x+i,k—i,0) n = 1, 2, 3 . . . . k. 



i=0 



In (5), setting n = 0, m = 1, and t+1 for /: 



f(t+l,x,k,0) =f(t,x,k,l) +xf(t,x > k,0) 



