FT6: Trigonometric-Style Identities for the Fibonacci and Lucas Numbers

This is the sixth 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

 

The Fibonacci numbers and Lucas numbers obey many identities analogous to the trigonometric and hyperbolic identities, with F playing the role of sin or sinh, and L playing the role of cos or cosh. These can be proven either via the closed-form expressions or by mathematical induction.

For example, analogous to the Pythagorean formula sin²x + cos²x = 1, we have:

• L(x)² – 5 F(x)² = 4 (–1)^x

Analogous to the double-angle formulas sin 2x = 2 sin x cos x and cos 2x = cos²x - sin²x, we have:

• F(2x) = F(x) L(x)
• 2 L(2x) = 5 F(x)² + L(x)²

There are addition formulas, corresponding to the standard trig formulas sin(x + y) = sin x cos y + cos x sin y and cos(x + y) = cos x cos y - sin x sin y:

• 2 F(x + y) = F(x) L(y) + F(y) L(x)
• 2 L(x + y) = 5 F(x) F(y) + L(x) L(y).

Unlike sine and cosine, the functions F and L aren't odd or even, but they do have opposite sign behavior in the following sense:

• F(–x) = (–1)^x+1 F(x)
• L(–x) = (–1)^x L(x).

You can combine the above to get subtraction formulas:

• 2 F(x – y) = (-1)^y (F(x) L(y) – F(y) L(x))
• 2 L(x – y) = (-1)^y (L(x) L(y) – 5 F(x) F(y))

Product formulas can be derived from the addition and subtraction formulas:

• 5 F(x) F(y) = L(x + y) + (–1)^y+1L(x – y)
• L(x) L(y) = L(x + y) + (–1)^yL(x – y)
• F(x) L(y) = F(x + y) + (–1)^yF(x – y)

Here are a couple of formulas just involving the traditional Fibonacci numbers:

• F(2x)² = F(x)² (5 F(x)² + 4 (-1)^x)
• F(3x) = F(x) (5 F(x)² + 3 (-1)^x)

... and just involving the Lucas numbers:

• L(2x) = L(x)² + 2 (-1)^x+1
• L(3x) = L(x) (L(x)² + 3 (-1)^x+1)

By the way, the formulas for F(2x), F(3x), and L(3x) show that F(x) divides F(2x) and F(3x), and that L(x) divides L(3x). We'll see later on that these are specific examples of general divisibility properties.