Lucas Polynomials

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

:$F_n(x)= \begin 0, & \mbox n = 0\\ 1, & \mbox n = 1\\ x F_(x) + F_(x),& \mbox n \geq 2 \end$

The Lucas polynomials use the same recurrence with different starting values:

:$L_n(x) = \begin 2, & \mbox n = 0 \\ x, & \mbox n = 1 \\ x L_(x) + L_(x), & \mbox n \geq 2. \end$

They can be defined for negative indices bySpringer
:$F_(x)=(-1)^F_(x),$
:$L_(x)=(-1)^nL_(x).$

The Fibonacci polynomials form a sequence of orthogonal polynomials with $A_n=C_n=1$ and $B_n=0$.

Examples

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 first few Lucas polynomials are:
:$L_0(x)=2 \,$
:$L_1(x)=x \,$
:$L_2(x)=x^2+2 \,$
:$L_3(x)=x^3+3x \,$
:$L_4(x)=x^4+4x^2+2 \,$
:$L_5(x)=x^5+5x^3+5x \,$
:$L_6(x)=x^6+6x^4+9x^2 + 2. \,$

Properties

* The degree of ''F''_''n'' is ''n'' − 1 and the degree of ''L''_''n'' is ''n''.

* The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at ''x'' = 1; Pell numbers are recovered by evaluating ''F''_''n'' at ''x'' = 2.

* The ordinary generating functions for the sequences are:
*:$\sum_^\infty F_n(x) t^n = \frac$
*:$\sum_^\infty L_n(x) t^n = \frac.$

*The polynomials can be expressed in terms of Lucas sequences as
*:$F_n(x) = U_n(x,-1),\,$
*:$L_n(x) = V_n(x,-1).\,$

*They can also be expressed in terms of Chebyshev polynomials $\mathcal_n(x)$ and $\mathcal_n(x)$ as
*:$F_n(x) = i^\cdot\mathcal_(\tfrac2),\,$
*:$L_n(x) = 2\cdot i^n\cdot\mathcal_n(\tfrac2),\,$
:where $i$ is the imaginary unit.

Identities

As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as
:$F_(x)=F_(x)F_n(x)+F_m(x)F_(x)\,$
:$L_(x)=L_m(x)L_n(x)-(-1)^nL_(x)\,$
:$F_(x)F_(x)- F_n(x)^2=(-1)^n\,$
:$F_(x)=F_n(x)L_n(x).\,$
Closed form expressions, similar to Binet's formula are:
:$F_n(x)=\frac,\,L_n(x)=\alpha(x)^n+\beta(x)^n,$
where
:$\alpha(x)=\frac,\,\beta(x)=\frac$
are the solutions (in ''t'') of
:$t^2-xt-1=0.\,$
For Lucas Polynomials ''n'' > 0, we have
:$L_n(x)=\sum_^ \frac \binom x^.$

A relationship between the Fibonacci polynomials and the standard basis polynomials is given byA proof starts from page 5 i
Algebra Solutions Packet (no author)

:x^n=F_(x)+\sum_^(-1)^k\left binom nk-\binom n\right_(x).
For example,
:$x^4 = F_5(x)-3F_3(x)+2F_1(x)\,$
:$x^5 = F_6(x)-4F_4(x)+5F_2(x)\,$
:$x^6 = F_7(x)-5F_5(x)+9F_3(x)-5F_1(x)\,$
:$x^7 = F_8(x)-6F_6(x)+14F_4(x)-14F_2(x)\,$

Combinatorial interpretation

If ''F''(''n'',''k'') is the coefficient of ''x^k'' in ''F_n''(''x''), namely
:$F_n(x)=\sum_^n F(n,k)x^k,\,$
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. 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)=\begin\displaystyle\binom &\textn \not\equiv k \pmod 2,\\$2pt0 &\text.
\end

This gives a way of reading the coefficients from Pascal's triangle as shown on the right.

References

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*Jin, Z. On the Lucas polynomials and some of their new identities. Advances in Differential Equations 2018, 126 (2018). https://doi.org/10.1186/s13662-018-1527-9

Further reading

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External links

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*{{OEIS el, sequencenumber=A011973, name=Triangle of coefficients of Fibonacci polynomials, formalname=Triangle of numbers {C(n-k,k), n >= 0, 0 <= k <= floor(n/2)}; or, triangle of coefficients of (one version of) Fibonacci polynomials

Polynomials
Fibonacci numbers