FINITE DIFFERENCES AND LINEAR DIFFERENTIAL EQUATIONS. 
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whence 
jo + 2)(a?— ^ + 1) {x—l){x—2)} x(^x—p + l 57 — 2 ))"^”} 
from which the series for can be expressed. 
Ex. 3. Let the equation be 
~\~XU^_2‘ 
Then 
K.=c<f|l+i2'(*+l)+i2'((.r+l)2'-'(^+l)) + ...J 
= Ja’+^x{x+l)a’-'+~{x+l)x{x—l){x—2)a’-‘+^^(x+l)x...(x—4)a’-‘>+.^. 
Ex. 4. Let the equation be 
I 2 
U^ = -U^-1 + CU^_2. 
Then 
Ur = 
l+^S'4x+l)+^2'(4*+l)S'-'*(*+l))+..} 
'*“r(a? + l) 
71^ f 1 
=I>+I)p + ? #- 1M*+ 1) 
+^(|(*+l)^(*— 1)(*— 2)(*— 3)(x— 4)+|(x+l)*(a;— 1)(®— 2)(a:— 3))+..| 
Ex. 5. Let the equation be 
Here 
N' -1 N" 
N" 
2 2 
{x + l)x{x—l){x—2) (jo + 2)(j9 + l) (a^+l)a? (p + 4)(jo + 3)(;) + 2)(p + l) 
XfP 
2.4 
2.4 
_(^ + l)ir(«— l)(i2?— 2)(.r— 3)(a?— 4) (^ + 2)(p + l) (i2? + l)ij;'(a? — l)(a? — 2) 
~ 2.4.6 “ 2 ' 2.4 
(p + 4)(jo + 3)(;? + 2)(jo+l) (a; + l)a? (p + 6)(p + 5)(;)4-4)(^ + 3)(jo + 2)(;) + l) 
2.4 ‘2 2.4.6 ’ 
-|^{m+2) 2»i) (jo + 2)(jt3 + l) (^ + !)..(;»— 2m + 2) (jj + 4)..(;) + 1) (a?+ 1). •{;*■ — 2m + 4) 
~ 2.4..(2m + 2) 2 ‘ 2.4. .2m 2.4 2.4.. (2m — 2) 
, ■ (jP + 2m)..(jo + l) (a;+l)a; -^ (jo + 2m + 2)..(j9+l) 
2.4. .2m ‘ 2 2.4..(2m + 2) 
