where Cn+1 =un+1vn+2 −un+2vn+1 is called the Casoratian (the analog of the Wron-
skian for continuous systems). It can be shown that Cn+1 is never zero if un, vn are
linearly independent solutions of equation (10.60). The first order difference equa-
tions are a special case of problem 21 of the exercises at the end of this chapter,
where it is demonstrated that the solutions can be written in the form
αn=α 0 −
n∑− 1
i=0
fivi+1
Ci+1
βn=β 0 +
n∑− 1
i=0
fiui+1
Ci+1
(10 .67)
and the general solution to equation (10.60) can be expressed as
yn=α 0 un+β 0 vn−un
n∑− 1
i=0
fivi+1
Ci+1 +vn
n∑− 1
i=0
fiui+1
Ci+1. (10 .68)
Analogous to the nth-order linear differential equation with constant coefficients
L(D) = (Dn+a 1 Dn−^1 +···+an− 1 D+an)y=r(x) (10 .69)
with D=dxd a differential operator, there is the nth-order difference equation
L(E) = (En+a 1 En−^1 +···+an− 1 E+an)yk=r(k), (10 .70)
where E is the stepping operator satisfying Ey k=yk+1. Most theorems and tech-
niques which can be applied to the ordinary differential equation (10.69) have anal-
ogous results applicable to the difference equation (10.70).
