Fibonacci polynomials explained

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:^[1]

F _{n(x)=} \begin{cases} 0 , & if n = 0 \\ 1 , & if n = 1 \\ x F _{n} (x) + F _{n} (x),& if n \geq 2 \end{cases}

The Lucas polynomials use the same recurrence with different starting values:^[2]

L _{n(x)} = \begin{cases} 2 , & if n = 0 \\ x, & if n = 1 \\ x L _{n} (x) + L _{n} (x), & if n \geq 2 . \end{cases}

They can be defined for negative indices by^[3]

F _{-n} (x) = ( -1 ) ^{n-1} F _{n} (x), L _{-n} (x) = ( -1 ) ^{nL} _{n} (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 ^{2+1} _{3 (x) = x} F ^{3+2 x} _{4 (x) = x} F ^{4+3 x} _{5 (x) = x} ^{2+1} F ^{5+4 x} _{6 (x) = x} ^{3+3x}

The first few Lucas polynomials are:

L _{0(x)=2} L _{1(x)=x} L ^{2+2} _{2 (x) = x} L ^{3+3 x} _{3 (x) = x} L ^{4+4 x} _{4 (x) = x} ^{2+2} L ^{5+5 x} _{5 (x) = x} ^{3+5x} L ^{6+6 x} _{6 (x) = x} ^{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 ^{inf ty} _{n =0} F _{n(x)} t ^{n} = (t)/(1- xt - t ^{2}) \sum ^{inf ty} _{n =0} L _{n(x)} t ^{n} = (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).} l{T} _{n(x)} and l{U} _{n(x)} asF _{n(x)} = i ^{n-1} ⋅ l{U} _{n-1} (\tfrac{ - ix} 2 ), L _{n(x)} = 2 ⋅ i ^{n ⋅ l{T}} _{n(\tfrac{ - ix} 2 ),}

where

i is the imaginary unit.

Identities

See main article: Lucas sequence. As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as

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} ^{nL} ^{m - n} (x) F _{n+1} (x)F _{n-1} (x) - F ^{2= ( -1 )} _{n(x)} ^{n} F _{2n} (x) = F _{n(x)L} _{n(x).} Closed form expressions, similar to Binet's formula are:F _{n(x) = (\alpha(x) ^{n-\beta(x)} ^{n})/(\alpha(x) - \beta(x))} ,L ^{n + \beta(x)} _{n(x) = \alpha(x)} ^{n,} where \alpha(x) = (x + \sqrt{x ^{2+4})/() },\,\beta(x)=\fracare the solutions (in t) oft ^{2-xt-1=0.} For Lucas Polynomials n > 0, we haveL _{n(x)=\sum} ^{\lfloor n/ 2 \rfloor} ^{k =0} (n)/(n - k) \binom{n - k}{k} x ^{n-2k} .

A relationship between the Fibonacci polynomials and the standard basis polynomials is given by^[4]

x ^{n = F} _{n +1} (x) + \sum ^{\lfloor n/ 2 \rfloor} _{k =1} ( -1 ) ^{k\left[\binom} nk - \binom n{k -1 }\right]F _{n+1-2k} (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} ^{k =0} 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) = \begin{cases}\displaystyle\binom{ (12 (n + k -1 )}{k})/(& if n) \not\equiv k \pmod 2 ,\\[ 12 pt] 0 & else . \end{cases}

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

References

• Book: Proofs that Really Count: The Art of Combinatorial Proof . Arthur T. . Benjamin . Arthur T. Benjamin . Jennifer J. . Quinn . Jennifer Quinn . . Dolciani Mathematical Expositions . 27 . 2003 . 978-0-88385-333-7 . Fibonacci and Lucas Polynomial . 141 .

• • • • 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

• V. E.. Hoggatt . Verner Emil Hoggatt, Jr. . Marjorie . Bicknell . Roots of Fibonacci polynomials. . . 11 . 271–274 . 1973 . 0015-0517. 0332645 .
• V. E.. Hoggatt . Calvin T. . Long . Divisibility properties of generalized Fibonacci Polynomials . . 12 . 113 . 1974 . 0352034 .
• Ricci . Paolo Emilio . Generalized Lucas polynomials and Fibonacci polynomials . Rivista di Matematica della Università di Parma. V. Ser. . 4 . 1995 . 137–146 . 1395332 .
• Yuan . Yi . Zhang. Wenpeng . Fibonacci Quarterly. 2002 . Some identities involving the Fibonacci Polynomials . 314. 1920571. 40. 4.
• Johann. Cigler . Fibonacci Quarterly . 2003 . 1962279 . 31–40. 41. q-Fibonacci polynomials.

Notes and References

1. Benjamin & Quinn p. 141
2. Benjamin & Quinn p. 142
3. Springer
4. A proof starts from page 5 in Algebra Solutions Packet (no author).