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. ==definition== these fibonacci polynomials are defined by a recurrence relation:benjamin & quinn p. 141 :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:benjamin & quinn p. 142 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&nbsp;=&nbsp;1; pell numbers are recovered by evaluating f'n at x&nbsp;=&nbsp;2. the degrees of f'n is n&nbsp;&minus;&nbsp;1 and the degree of l'n is n. the ordinary generating function for the sequences are: : \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).\, ==identities==  as particular cases of lucas sequences, fibonacci polynomials satisfy a number of identities. first, they can be defined for negative indices byspringer :f_{-n}(x)=(-1)^{n-1}f_{n}(x),\,l_{-n}(x)=(-1)^nl_{n}(x). other identities include: :f_{m+n}(x)=f_{m+1}(x)f_n(x)+f_m(x)f_{n-1}(x)\, :l_{m+n}(x)=l_m(x)l_n(x)-(-1)^nl_{m-n}(x)\, :f_{n+1}(x)f_{n-1}(x)- f_n(x)^2=(-1)^n\, :f_{2n}(x)=f_n(x)l_n(x).\, closed form expressions, similar to binet's formula are: :f_n(x)=\frac{\alpha(x)^n-\beta(x)^n}{\alpha(x)-\beta(x)},\,l_n(x)=\alpha(x)^n+\beta(x)^n, where :\alpha(x)=\frac{x+\sqrt{x^2+4}}{2},\,\beta(x)=\frac{x-\sqrt{x^2+4}}{2} are the solutions (in t) of :t^2-xt-1=0.\, ==combinatorial interpretation== thumb|right|360px|the coefficients of the fibonacci polynomials can be read off from pascal's triangle following the "shallow" diagonals (shown in red). the sums of the coefficients are the fibonacci numbers. 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 nâ1 by 1 rectangle can
Fibonacci_numbers Polynomials