Recurrence Resolution Path

Lesson 2: From Recursive Definitions to Closed Forms

The Efficiency Problem

The Recursive Way

\( F_{1000} = F_{999} + F_{998} \)

(Requires 999 previous steps)

The Closed-Form Way

\( F_n = \frac{1}{\sqrt{5}} \left( \phi^n - \psi^n \right) \)

(Calculate immediately for any \( n \))

The Method of Generating Functions

1. Define the Series

Write \( A(x) = \sum_{n=0}^{\infty} a_n x^n \).

2. Multiply and Sum

Apply the recurrence to the sum over \( n \).

3. Solve for \( A(x) \)

Isolate \( A(x) \) as a rational function.

4. Extract Coefficients

Use Partial Fractions to find \( a_n \).

Visualizing Step 2

\( a_n = c_1 a_{n-1} + c_2 a_{n-2} \)
\( \downarrow \)
\( \sum a_n x^n = c_1 \sum a_{n-1}x^n + \dots \)

The Fibonacci Derivation

Algebraic Setup

\( A(x) = x + x A(x) + x^2 A(x) \)
\( A(x)(1 - x - x^2) = x \)
\( A(x) = \frac{x}{1 - x - x^2} \)

Denominator Roots

\( 1-x-x^2 = (1-\phi x)(1-\psi x) \)
Where \( \phi = \frac{1+\sqrt{5}}{2} \) and \( \psi = \frac{1-\sqrt{5}}{2} \)

Partial Fraction Result:

\( \frac{x}{(1-\phi x)(1-\psi x)} = \frac{1}{\sqrt{5}} \left( \frac{1}{1-\phi x} - \frac{1}{1-\psi x} \right) \)

The Magic of Linearity

The denominator of the GF is the characteristic polynomial in disguise.

If the recurrence is \( a_n = c_1 a_{n-1} + c_2 a_{n-2} \), then the GF denominator is \( 1 - c_1 x - c_2 x^2 \).

Roots of the polynomial give the basis functions of the sequence.

Takeaway

Generating functions solve recurrences not by "guessing and checking," but by performing a basis transformation in the ring of power series.

Combinatorial Architectures

Formal Series Foundations

Recurrence Resolution Path

Geometric Counting Logic

Labeled Structure EGFs

Snake Oil Mastery

Formal Series Slides

Formal Foundations Handout

Series Architect Teacher Guide

Recurrence Resolver Slides

Fibonacci Formula Worksheet

Partial Fraction Master Key

Combinatorial Coin Slides

Change Maker Activity

Counting Constraints Teacher Guide

Labeled Structures Slides

Permutation Power Worksheet

EGF Expert Answer Key

Snake Oil Slides

Identity Investigator Worksheet

Mastering Sums Teacher Guide

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