Mathematics | Generating Functions – Set 2 Last Updated : 29 May, 2021 Improve Improve Like Article Like Save Share Report 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. GATE CS 2018 | Question 18GATE-CS-2017 (Set 2) | Question 52 Like Article Suggest improvement Previous Discrete Maths | Generating Functions-Introduction and Prerequisites Next Mathematics | Sequence, Series and Summations Share your thoughts in the comments Add Your Comment Please Login to comment...