Lecture 21- More on Recurrence Equations

You may want to download the lecture slides that were used for these videos (PDF).

1. Vector Spaces and Linear Recurrence Equations

This video reviews material discussed in the previous lecture: in particular, we review vector spaces of complex functions and linear recurrence equations. (3:11)

2. The Advancement Operator

This video reviews material discussed in the previous lecture: in particular, we discuss rewriting linear homogeneous equations in a polynomial form using the advancement operator. (1:02)

3. The General Theorem

This video reviews the theorem that we proved at the end of the previous lecture, describing the solution space for a homogeneous linear recurrence equation.  Then, it explores an example where we characterize the general solution to an advancement operator equation. (6:01)

4. Using Initial Conditions

This video describes how to use initial conditions to reduce the general solution of a recurrence equation to a solution appropriate for a given problem. (4:19)

5. When Zero is a Root

In this video we consider the equation Amf(n) = 0. (3:39)

6. The Non-Homogeneous Case

Now we consider the case p(A)f = g. We introduce and prove a theorem that gives the general form of the solution. (7:16)

7. Proof of the General Theorem

Here we restate a theorem that characterizes the solution space to an operator equation.  We will prove the theorem in subsequent videos. (1:00)

8. A Key Lemma

This video states a key lemma and describes why the lemma works.  For any polynomial p(n) and any real number r, (A – r)p(n)r^n = q(n) r^n, where q is a polynomial with degree less than the degree of p.  (9:51)

9. A Useful Corollary

We state and discuss a corollary of our lemma. (1:31)

10. Repeated Roots

This video provides a theorem about the general solution to a recurrence equation with repeated roots.  Then, we state how we will prove it in subsequent videos. (2:51)

11. First and Second Parts

This video gives the first and second parts of our proof. (5:27)

12. The Third Part – Linear Independence

This video gives the third part of our proof, which deals with linear independence. (3:40)

13. Multiple Roots

We’ve so far assumed that we have a single root. In this video we extend our results to the case when we have multiple roots. (3:36)

14. Fibonacci Numbers

We use the concepts we explored in this lecture to the Fibonacci sequence. (12:54)