445 
$4. 
The properties of 
i k 
= i(k -\n . 
finye,k) = = (1) (8) @t4) §2 
i=0 
: i(k) 1 
and an application of > (—1)' 7] Parr in the theory ot gamma functions 
i=0 : 
suggests the generalization: 
ke . (k) 
(1) f(tx,kyn) = (1) |} @+0 
i=0 
FT ODES on eet Qu Ieee 8 
Whence 
7 0.7,k,n) = S(kn) eyes (seller ers care Mad S ichycs ceases 
(3) f(t,z,0,n) = a! when n = 
=a) when n > 0 
(4) Fier) a — ao (x+1)! when n = 0 
- —=(¢-E1) when n > 0 
When ¢ < 0, this function has poles at x = —1,—2,...... —k, and 
also whenn + ¢ < 0, atx = 0. 
LS a 
Since f(t,x,kyn) = & (—1)' [yy oti ™ ti)” 
i=0 
we have the recursion formula 
ls ( : 
(5) f(t,r,k,n) = > i ax f(t—m,z,k,m+n—1) 
i=0 
Oe rans cute ea el aeons) ee aan cai CUTE — Lacy thee oceans 
If t is not negative, we have on setting ¢ for m in (5) 
t (ey 
(6) fli,z,kn) = > Li 2 S(t+n—i) kw i0 be 35. 
i=0 
lfe0<nck 
Pee (el 
2» (1) La a” Gti = Aye ¥Ge4tnk—n, 0) n= 1, 253 ...2 
i=0 : 
Wh ence, making use of (2) §3, 
n 
(7) f(ta,kyn) = & (—1)' A(in) k” f(t,2-+4,k4,0) AA k 
i=0 
In (5), setting n = 0, m = 1, and t+1 for ft: 
S(t-+1,2,k,0) = f(t,x,k,1) + x f(t,x,k,0) 
