Advanced High-School Mathematics

SECTION 2.1 Elementary Number Theory 101

However, given thatu 0 = 0, u 1 = 1, we get

0 = A

1 = Acos

2 π

3

+Bsin

2 π

3

=−

A

2

+

√

3 B

2

ThereforeA= 0 andB=

√^2

3

, forcing the solution to be

un =

2

√

3

sin

2 πn

3

, n= 0, 1 , 2 ,....

Higher-degreecharacteristic polynomials.

We won’t treat this case systematically, except to say that upon

factoring the polynomial into irreducible linear and quadratic fac-

tors, then one can proceed as indicated above (see Exercise 14).

Additional complications result with higher-order repeated factors

which we don’t treat here.

Higher-order differences

In Section 2.1.9 we treated only the so-called homogeneous linear
difference equations. An inhomogeneous linear difference equation
has the general form

C(u) =v,

whereC(x) is a monic polynomial,v= (vn)n≥ 0 is a given sequence and
whereu= (un)n≥ 0 is the unknown sequence.

We have already encountered such an example above, in the example
on page 312 giving an arithmetic sequence:

un+1−un=d, n= 0, 1 , 2 ,....

We won’t treat inhomogeneous linear difference equations in any
detail except for a very special case, namely those having constant