FT2: The Canonical Basis & Closed-Form Expressions

This is the second in a series of posts on Fibonacci trigonometry:

Part 1: Basic Definitions

Part 2: The Canonical Basis and Closed-Form Expressions

Part 3: The Basic Analogies

Part 4: Fibonacci Formulas Using sinh, etc. — Real Number Version

Part 5: Fibonacci Formulas Using sinh, etc. — Complex Number Version

Part 6: Trigonometric-Style Identities

Part 7: The Fibonacci Tangent and Cotangent Functions

Part 8: Recurrence Relations for Ftan and Fcot

 

If u satisfies the equation 1 + u = u², then the function that maps n to uⁿ is a generalized Fibonacci sequence. We'll call this function u*.

There are two solutions to the equation 1 + u = u²: φ = (1 + √5)/2 = 1.618... (the "golden ratio"), and φ' = 1–φ = –1/φ = (1 – √5)/2 = -0.618....

One can see that any generalized Fibonacci function can be written in the form a φ* + b φ'* for uniquely determined constants a and b.

In other words, {φ*, φ'*} is a basis for the vector space of generalized Fibonacci functions (under the naturally defined pointwise operations).

This gives us a closed-form expression for any generalized Fibonacci function f: Take a = (–φ' f (0) + f (1))/√5 and b = (φ f (0) – f (1))/√5. Then f (n) = a φ* (n) + b φ'* (n) for every integer n.

For our purposes, it's useful to rewrite a φ*(n) + b φ'*(n) as a φⁿ + b (–φ)^–n.

In particular, the standard Fibonacci and Lucas sequences can be written as follows:

• F(n) = (φⁿ – (–φ)^–n)/√5
• L(n) = φⁿ + (–φ)^–n

Note that I haven't tried to normalize these in any sense, since I wanted to keep the standard integer-valued functions.