Mathematics | Generating Functions - Set 2

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 repetitive 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 containingis .For RHS the term containingis . So (proved) 

Links of Various examples are given below regarding generating functions. 
 

1. GATE CS 2018 | Question 18
2. GATE-CS-2017 (Set 2) | Question 52