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DISCRETE MATHEMATICS
W W L CHEN
c
±
W W L Chen, 1992, 2008.
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Chapter 16
RECURRENCE RELATIONS
16.1. Introduction
Any equation involving several terms of a sequence is called a recurrence relation. We shall think of the
integer
n
as the independent variable, and restrict our attention to real sequences, so that the sequence
a
n
is considered as a function of the type
f
:
N
∪ {
0
} →
R
:
n
7→
a
n
.
A recurrence relation is then an equation of the type
F
(
n,a
n
,a
n
+1
,...,a
n
+
k
) = 0
,
where
k
∈
N
is ﬁxed.
Example 16.1.1.
a
n
+1
= 5
a
n
is a recurrence relation of order 1.
Example 16.1.2.
a
4
n
+1
+
a
5
n
=
n
is a recurrence relation of order 1.
Example 16.1.3.
a
n
+3
+ 5
a
n
+2
+ 4
a
n
+1
+
a
n
= cos
n
is a recurrence relation of order 3.
Example 16.1.4.
a
n
+2
+ 5(
a
2
n
+1
+
a
n
)
1
/
3
= 0 is a recurrence relation of order 2.
We now deﬁne the order of a recurrence relation.
Definition.
The order of a recurrence relation is the diﬀerence between the greatest and lowest sub
scripts of the terms of the sequence in the equation.
Definition.
A recurrence relation of order
k
is said to be linear if it is linear in
a
n
,a
n
+1
,...,a
n
+
k
.
Otherwise, the recurrence relation is said to be nonlinear.
Chapter 16 : Recurrence Relations
page 1 of 17
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c
±
W W L Chen, 1992, 2008
Example 16.1.5.
The recurrence relations in Examples 16.1.1 and 16.1.3 are linear, while those in
Examples 16.1.2 and 16.1.4 are nonlinear.
Example 16.1.6.
a
n
+1
a
n
+2
= 5
a
n
is a nonlinear recurrence relation of order 2.
Remark.
The recurrence relation
a
n
+3
+ 5
a
n
+2
+ 4
a
n
+1
+
a
n
= cos
n
can also be written in the form
a
n
+2
+5
a
n
+1
+4
a
n
+
a
n

1
= cos(
n

1). There is no reason why the term of the sequence in the equation
with the lowest subscript should always have subscript
n
.
For the sake of uniformity and convenience, we shall in this chapter always follow the convention that
the term of the sequence in the equation with the lowest subscript has subscript
n
.
16.2. How Recurrence Relations Arise
We shall ﬁrst of all consider a few examples. Do not worry about the details.
Example 16.2.1.
Consider the equation
a
n
=
A
(
n
!), where
A
is a constant. Replacing
n
by (
n
+ 1) in
the equation, we obtain
a
n
+1
=
A
((
n
+1)!). Combining the two equations and eliminating
A
, we obtain
the ﬁrstorder recurrence relation
a
n
+1
= (
n
+ 1)
a
n
.
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This note was uploaded on 12/20/2009 for the course MATH 245 taught by Professor ramaswamy during the Spring '08 term at San Diego State.
 Spring '08
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 Math

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