Homogeneous Recurrence Relation In Discrete Mathematics Pdf
The manner in which the terms of a sequence are found in recursive manner is called recurrence relation. An equation which defines a sequence recursively, where the next term is a function of the previous terms is known as recurrence relation. A recurrence relation is an equation that recursively defines a sequence A linear recurrence equation of degree k or order k is a recurrence equation which is in the format (An is a constant and Ak≠0) on a sequence of numbers as a first-degree polynomial. Some of the examples of linear recurrence equations are as follows: Recurrence relations Initial values Solutions Fibonacci number Lucas Number Padovan sequence Pell number For instance, the two ordered linear recurrence relation is - where A and B are real numbers. For the above recurrence relation, the characteristic equation is : Solve the recurrence relation Solution For the recurrence relation, the characteristic equation is: Solving these two equations, we get a=2 and b=−1 Hence, the final solution is − For the recurrence relation, the characteristic equation is: Solve the recurrence relation For the recurrence relation, the characteristic equation is as follows: The roots are imaginary. So, this is in the form of case 3. Hence, the solution is − Solving these two equations we get a=1 and b=2 Hence, the final solution is − A non-homogeneous recurrence is in the form of: The associated homogeneous recurrence relation will be There are two parts of a solution of a non-homogeneous recurrence relation. The first part of the solution is the solution of the associated homogeneous recurrence relation and the second part of the solution is the solution of that particular solution . By finding an appropriate trial solution, the particular solution can be found. For instance, the characteristic equation of the associated homogeneous recurrence relation be For the non-homogeneous recurrence relation With the characteristic roots of The trial solution is as follows: For the linear non-homogeneous relation, the associated homogeneous equation is: The characteristic equation of its associated homogeneous relation is − Once the solution is put in the recurrence relation, the result obtained is: The solution of the recurrence relation can be written as − When each term of a sequence is expressed as a coefficient of the variable x in a power series, the sequence is represented as Generating functions. Mathematically, for an infinite sequence, say a0,a1,a2,…,ak,…,a0,a1,a2,…,ak,…, the generating function will be − Generating functions can be used for the following purposes − What are the generating functions for the sequences Solution For the infinite series - 1,1,1,1,…, what is the generating function? Solution What is Recurrence Relation in Discrete Mathematics?
Definition
What is Linear Recurrence Relations?
How to solve linear recurrence relation?
Problem 1
Problem 2
Solution
Problem 3
Solution
Non-Homogeneous Recurrence Relation and Particular Solutions
Example
Problem
Solution
What is Generating Functions?
Some Areas of Application of Generating Functions
Problem 1
Problem 2
What are the different useful generating functions?
Homogeneous Recurrence Relation In Discrete Mathematics Pdf
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