Fibonacci numbers and orthogonal polynomials

The famous Fibonacci numbers are defined by the initial conditions

F0 = 0,  F1 = 1

and the recurrence relation

Fn = Fn-1 + Fn-2

for n > 1.

The Fibonacci polynomials are defined similarly. The have the same initial conditions

F0(x) = 0, F1(x) = 1

but a slightly different recurrence relation

Fn(x) = x Fn-1(x) + Fn-2(x)

for n > 1.

Several families of orthogonal polynomials satisfy a similar recurrence relationship

Fn(x) = p(x) Fn-1(x) + c(n) Fn-2(x).

The table below gives the values of p(x) and c(n) for a few families of polynomials.

Family p(x) c(n) P0 P1(x)
Fibonacci x 1 0 1
Chebyshev T 2x -1 1 x
Chebyshev U 2x -1 1 2x
Hermite 2x 2 – 2n 1 2x

There are two common definitions of the Hermite polynomials, sometimes called the physicist convention and the probabilist convention. The table above gives the physicist convention. For the probabilist definition, change the 2’s to 1’s in the last row and leave the 1 alone.

(If you haven’t heard of orthogonal polynomials, you might be wondering “orthogonal to what?”. These polynomials are orthogonal in the sense of an inner product formed by multiplying two polynomials and a weight, then integrating. Orthogonal polynomials differ by the the weight and the limits of integration. For example, the (physicist) Hermite polynomials are orthogonal with weight function exp(-x2) and integrating over the entire real line. The probabilist Hermite polynomials are very similar, but use the standard normal distribution density exp(-x2/2)/√(2π) as the weight instead of exp(-x2).)

2 thoughts on “Fibonacci numbers and orthogonal polynomials

  1. More generally, EVERY family of orthogonal polynomials satisfy the property: F_{n+2}(x) = (a_n x + b_n) F_{n+1}(x) + c_n F_{n}(x), F_1 = const, F_2 = a x + b, a_n 0, c_n 0. The only requirement: the weight must be non-negative (non-zero function). Symmetric families (integration segment [-a, a], weight p(x) = p(-x)) have b_n = 0.

  2. Simple introduction to orthogonal polynomials, use of recurrence relations. More valuable if there are specific MUST read references to the students referring this blog.

    Great in deed

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