Fibonacci polynomial

In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials.


These Fibonacci polynomials are defined by a recurrence relation:[1]

F_n(x)= \begin{cases}

0, & \mbox{if } n = 0\\ 1, & \mbox{if } n = 1\\ x F_{n - 1}(x) + F_{n - 2}(x),& \mbox{if } n \geq 2 \end{cases}

The first few Fibonacci polynomials are:

F_0(x)=0 \,
F_1(x)=1 \,
F_2(x)=x \,
F_3(x)=x^2+1 \,
F_4(x)=x^3+2x \,
F_5(x)=x^4+3x^2+1 \,
F_6(x)=x^5+4x^3+3x \,

The Lucas polynomials use the same recurrence with different starting values:[2] L_n(x) = \begin{cases} 2, & \mbox{if } n = 0 \\ x, & \mbox{if } n = 1 \\ x L_{n - 1}(x) + L_{n - 2}(x), & \mbox{if } n \geq 2. \end{cases}

The first few Lucas polynomials are:

L_0(x)=2 \,
L_1(x)=x \,
L_2(x)=x^2+2 \,
L_3(x)=x^3+3x \,
L_4(x)=x^4+4x^2+2 \,
L_5(x)=x^5+5x^3+5x \,
L_6(x)=x^6+6x^4+9x^2 + 2. \,

The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at x = 1; Pell numbers are recovered by evaluating Fn at x = 2. The degrees of Fn is n − 1 and the degree of Ln is n. The ordinary generating function for the sequences are:[3]

\sum_{n=0}^\infty F_n(x) t^n = \frac{t}{1-xt-t^2}.
\sum_{n=0}^\infty L_n(x) t^n = \frac{2-xt}{1-xt-t^2}.

The polynomials can be expressed in terms of Lucas sequences as

F_n(x) = U_n(x,-1),\,
L_n(x) = V_n(x,-1).\,


Main article: Lucas sequence

As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities.

First, they can be defined for negative indices by[4]


Other identities include:[4]

F_{n+1}(x)F_{n-1}(x)- F_n(x)^2=(-1)^n\,

Closed form expressions, similar to Binet's formula are:[4]




are the solutions (in t) of


Combinatorial interpretation

If F(n,k) is the coefficient of xk in Fn(x), so

F_n(x)=\sum_{k=0}^n F(n,k)x^k,\,

then F(n,k) is the number of ways an n−1 by 1 rectangle can be tiled with 2 by 1 dominoes and 1 by 1 squares so that exactly k squares are used.[1] Equivalently, F(n,k) is the number of ways of writing n−1 as an ordered sum involving only 1 and 2, so that 1 is used exactly k times. For example F(6,3)=4 and 5 can be written in 4 ways, 1+1+1+2, 1+1+2+1, 1+2+1+1, 2+1+1+1, as a sum involving only 1 and 2 with 1 used 3 times. By counting the number of times 1 and 2 are both used in such a sum, it is evident that F(n,k) is equal to the binomial coefficient


when n and k have opposite parity. This gives a way of reading the coefficients from Pascal's triangle as shown on the right.


  • MathWorld.

Further reading

External links

  • "On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  • "On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
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