FT4: Formulas using Real Hyperbolic Functions

This is the fourth 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

 

So we've seen that the Fibonacci and Lucas sequences have properties analogous to sin and cos, or sinh and cosh. But does this go beyond an analogy to an actual connection? Can we write F in terms of sin or sinh, etc.?

Yes. Here's one way...

We'll write these expressions using hyperbolic functions.

F(n) =

• 2 sinh (n ln φ) / √5, if n is even,
• 2 cosh (n ln φ) / √5, if n is odd.

L(n) =

• 2 cosh (n ln φ), if n is even,
• 2 sinh (n ln φ), if n is odd.

Ftan(n) =

• 2 tanh (n ln φ) / √5, if n is even,
• 2 coth (n ln φ) / √5, if n is odd.

Fcot(n) =

• √5 coth (n ln φ) / 2, if n is even,
• √5 tanh (n ln φ) / 2, if n is odd.

Fsec(n) =

• sech (n ln φ) / 2, if n is even,
• csch (n ln φ) / 2, if n is odd.

Fcsc(n) =

• √5 csch (n ln φ) / 2, if n is even,
• √5 sech (n ln φ) / 2, if n is odd.

In the next section, we'll see how we can use complex numbers in the formulas to avoid splitting into even and odd cases.