Lecture Videos Lecture 21- More on Recurrence Equations

# 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)

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)