Fibonacci polynomials: F_1=1, F_2=x, F_{n+1}=xF_n+F_{n-1}. So F_3=x²+1, F_4=x³+2x, F_5=x⁴+3x²+1. At x=1: 1,1,2,3,5,8... (Fibonacci!). Coefficients are binomial-like. Related to Chebyshev: F_n(x)=i^{1-n}U_{n-1}(ix/2).
How to Use
Enter n and x:
- F_n(x): Fibonacci poly
- L_n(x): Lucas poly
- x=1: Classic numbers
Coefficient Pattern
F_n(x) = Σ C(n-1-k,k)·x^{n-1-2k}. Coefficients are entries from the 'shallow diagonal' of Pascal's triangle. This gives the beautiful connection: Fibonacci numbers = shallow diagonal sums of Pascal's triangle.
Chebyshev Connection
F_n(x) = i^{1-n}·U_{n-1}(ix/2) where U is Chebyshev second kind. This means Fibonacci polynomials are 'hyperbolic Chebyshev' polynomials — the same recurrence but with + instead of − in the argument.
Step-by-Step Instructions
- 1Enter n.
- 2Enter x.
- 3Compute F_n(x).
- 4Compute L_n(x).
- 5Try x=1 for classic.