Topics to be Covered

1
Topics to be Covered

• Generating Functions
• Recurrence Functions

2
Recurrence Relations

• Consider the following
• 1. start with 5
• 2. Given any term, add 3 to get the next
  term
• If we list the terms we get
• 5, 8, 11, 14, 17,

3
Recurrence Relations

• If we write the terms in this list as a1, a2,
  an, we get
• a1 5
• an an-1 3 where n 2
• OR
• an 1 n1
• an-1 3 n 2
• This is an example of a recurrence relation

4
Recurrence Relations

• A recurrence relation for the sequence a0, a1,
  a2, is an equation that relates an to certain
  of its predecessors a0, a1, ,an-1
• Initial conditions for the sequence a0, a1,
  ,an-1 are explicitly given values for a finite
  number of terms of the sequence

5
Example

• Write down the first five terms of the sequence
  defined by a1 1, ak1 3ak1 for k 1
• a1 1
• a2 3a11 3(1)1 4
• a3 3a21 3(4)1 13
• a4 40
• a5 121
• .
• .
• .
• an ?

6
Example

• In this case a formula for the nth term is as
  follows
• an (1/2)(3n -1)
• This is what is called a solution to the
  recurrence relation a0 1, ak1 3ak1 for k 1
• Generating functions may be used to solve
  recurrence relations

7
Example

• Solve the recurrence relation
• a0 1
• an 3an-1 , n1
• Solution,
• Consider the generating function
• f(x) a0a1x a2x2 anxn
• 3xf(x) 3a0x3a1x23an-1xn
• Subtracting

8
Example

• Subtracting
• f(x)-3xf(x) a0 (a1- 3a0)x (a2- 3a1)x2
  (an- 3an-1)xn
• Now Substituting a0 1, a1 3a2 ,, an
  3an-1
• (1-3x)f(x) 1
• f(x) 1 / (1-3x)
• Recall 1 1xx2xn
• 1-x
• So 1 13x(3x)2(3x)n
• 1-3x
• 13x9x23nxn

9
Example

• an is the coefficient of the xn term in f(x)
• an 3n
• is the solution of the recurrence relation
• 1 , n0
• an
• 3a , n1

10
Example

• Solve the recurrence relation an 2an-1 an-2,
  ngt1, given a0 3 and a1 -2
• Solution
• f(x) a0a1x a2x2 anxn
• 2xf(x) 2a0x2a1x2 2an-1xn
• x2f(x) a0x2 an-2xn
• Therefore,

11
Example

• f(x)-2xf(x)x2 f(x)
• a0(a1-2a0)x(a2-2a1a0)x2(an-2an-1an-2)xn
• 3-8x
• Since a0 3
• a1 -2
• an 2an-1an-2 0 for n gt1
• Now, f(x)-2xf(x)x2 f(x) f(x)(1-x)2 3-8x
• Therefore, f(x) 3-8x
• (1-x)2

12
Example

• Now recall
• 1 12x3x24x3(n1)xn
• (1-x)2
• So, f(x) (12x3x2(n1)xn)(3-8x)
• 32x7x212x33(n1) 8nxn
• 32x7x212x3-5n 3xn
• And so an 3- 5n is the solution to the
  recurrence relation.