FT3: The Basic Analogies

This is the third 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

 

In part 2, we gave the following closed-form expressions for the Fibonacci numbers and the Lucas numbers:

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

(Recall that φ represents the golden ratio (1 + √5)/2 = 1.618....)

Using the formulas above, we can derive the following:

• L(n) + √5 F(n) = 2 φⁿ
  
• L(n) – √5 F(n) = 2 (–φ)^–n

The four formulas above are reminiscent of standard formulas relating the trigonometric and hyperbolic functions to the exponential function:

F(n) = (φn – (–φ)–n)/√5 L(n) = φn + (–φ)–n L(n) + √5 F(n) = 2 φn L(n) – √5 F(n) = 2 (–φ)–n

sin x = (eix – e–ix)/2i cos x = (eix + e–ix)/2   cos x + i sin x = eix cos x – i sin x = e–ix

sinh x = (ex – e–x)/2 cosh x = (ex + e–x)/2 cosh x + sinh x = ex cosh x – sinh x = e–x

  Because of these similarities, the Fibonacci and Lucas numbers obey many identities similar to well-known trigonometric and hyperbolic identities, as we'll see.
In fact, as we'll see in part 5, there are formulas for F(n) and L(n) in terms of sinh and cosh: F(n) and L(n) are constant multiples of sinh(nψ) and cosh(nψ), respectively, with the proper choice of ψ.