Prerequisite – Generating Functions-Introduction and Prerequisites
In Set 1 we came to know basics about Generating Functions. Now we will discuss more details on Generating Functions and its applications.
Exponential Generating Functions –
Let e a sequence. Then its exponential generating function, denoted by
is given by,
Example 1:- Let {1, 1, 1…….} be a sequence . The generating function of the sequence is
( Here
=1 for all n )
Example 2:- Let be number of k permutation in an n- element set. Then the exponential generating function for the sequence
is
Exponential Generating Function is used to determine number of n-permutation of a set containing repeatative elements. We will see examples later on.
Using Generating Functions to Solve Recurrence Relations –
Linear homogeneous recurrence relations can be solved using generating function .We will take an example here to illustrate .
Example :- Solve the linear homogeneous recurrence equation .
Given =1 and
.
We use generating function to solve this problem. Let g(x) be the generating function of the sequence .
Hence g(x)=
So we get the following equations.
g(x)=
-5xg(x)=
=
Adding these 3 quantities we obtain
Now =0 for all n>1. So,
Or g(x)=
Now =(1-2x)(1-3x)
So, g(x)=
It is easy to see that
Now
And
So g(x)=
Since this is the generating function for the sequence We observe that
Thus we can solve recurrence equations using generating functions.
Proving Identities via Generating Functions –
Various identities also can also be proved using generating functions.Here we illustrate one of them.
Example: Prove that :
Here we use the generating function of the sequence i.e
.
Now,
For LHS the term containing is
.For RHS the term containing
is
. So
(proved)
Links of Various examples are given below regarding generating functions.
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