MATH F213
SOLVING RECURRENCE RELATIONS
BITS Pilani
Hyderabad Campus
Michael Alphonse
The topic to be covered
1. Formulation of Recursive Relations
Solving Methods :
1. Substitution Method
2. Using Generating Functions
3. The Method of Characteristic Roots
BITS Pilani, Hyderabad Campus
Formulation
1. Find a recurrence relation for the number of ways to
arrange flags on a flagpole of n feet tall using 4 types of
flags : red flags 2 feet high, white, blue and yellow flags
each 1 foot high.
Ans : Fn = Fn-2 + Fn-1
2. A student starts a chain letter by writing to 4 of his friends
and asking that each of them write to 4 others and so on.
Let an denote the number of letters written at the nth stage
of the chain letter. Find and solve a recurrence relation.
Assume a0 = 4.
3. Find a recurrence relation that counts the number of
subsets of {1, 2, …n} having no twp consecutive numbers.
BITS Pilani, Hyderabad Campus
Substitution Method
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
Solving using generating
functions
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
Outline of the method
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
This can be written as 𝐴 𝑥 − 𝑎0 − 𝑎1 𝑥 − 7𝑥 𝑎1 𝑥 + 𝑎2 𝑥 2 + … + 10𝑥 2 𝑎0 +
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
The Method of Characteristic
Roots
Recall how we have solved the recurrence relation
an – 9an-1 +26an-2 – 24an-3 = 0 for n ≥ 3.
Solution is an= c12n + c23n + c34n.
Note that A(X) = P(X)/Q(X)
Where Q(X) = 1 – 9X + 26X2 – 24X3 = (1-2X)(1-3X)(1-4X).
The roots of the polynomial Q X) are ½, 1/3, ¼.
Let C(t) = t3 Q(1/t).= (t-2)(t-3)(t-4)
The polynomial C(t) is called characteristic polynomial of the
given recurrence relation and its roots are called
characteristic roots.
BITS Pilani, Hyderabad Campus
If the recurrence relation is
an + C1an-1 + … + Ckan-k = 0 for n ≥ k, then the characteristic
polynomial of the gieven recurrence relation is
C(t) = tk + C1tk-1 + … +ck = tkQ(1/t)
Where Q(X) = 1 + C1X + …+Ck Xk
If C(t) factors as (t-a1)(t-a2) … t-ak) then
In the expression A(X) = P(X)/Q(X),
Q(X) factors as (1-a1X)(1-a2X) … 1-akX)
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
How to Solve if the characteristic
Roots have multiple roots.
First we shall see how to use generating function, then we
will see how to use the method of characteristic roots.
Recall the formula
1
=
1 − 𝑥)𝑛
=
∞
𝑟=0 𝐶
∞
𝑛
𝑥𝑘)
𝑘=0
𝑛 − 1 + 𝑟, 𝑟) 𝑥 𝑟
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus
BITS Pilani, Hyderabad Campus