E[X, - X;--] = — a)E[X,-1 -X-,]—a2 El X-2-X,-+] 
+ Ele, X,,] + b)Ele 1 X--]. (5.153) 
Another way, by Green’s function, 
X;= Sy Gale) =i Goer Gir Gri Gon eo mom G keer n 
y=0 
where G ; is expressed by Eqs. 5.128 to 5.131. Accordingly, 
Mees (Cy GG ens Gagne? o 0G Epa icre ¢ © 
= € 7+ (by — 41 )€nr-1 + [-42—41(b1—41)} Enr2+- - - 
+ [- ayGi2 — 4G} Gee itiigis (5.154) 
Using these relations from Eq. 5.153 gives 
r=0 when R(0) =—a,R(1)—a2R(2)+02+(b1-a) bio2 ~~ (5.155) 
r=1 when R(1) =—a,R(0)—a2R(1) + by0?2 (5.156) 
r 22 when R(r) = —a)R(r —1)—anR(r—2). Galsy)) 
In order to express the values of R(r) recursively, we have to solve for R(O) and 
R(1), using Eqs. 5.155 and 5.156 after substituting R(2) from Eq. 5.157. Then Eqs. 5.155 
and 5.156 will be 
(1 — a2) RO) +.a;(1 —a2)R(1) = (1 —ayb, +b?) 2 (5.158) 
a,R(0) + (1 + a2)R(1) = byo2. (5.159) 
Therefore 
(1—a,b, +b?) a,(1—-a2) 
b; (1 + a2) 
RO) = ———qqw— os ( 5.160) 
(1—a#) a,(1—a2) 
Qa (1 + a>) 
(1 —a#) (1 —a1b; +b?) 
aj by 
R() = ———————_- 02, (5.161) 
(1—a3) a,(1—a) 
Qa, (1 + a2) 
R(r) = —a\R(r—1)-a2R(r-2) or = 2, (5.162) 
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