Fibonacci polynomials

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[Triangle of] Fibonacci polynomials

F0 (x)	0
F1 (x)	1
F2 (x)	1
F3 (x)	1 x + 1
F4 (x)	2 x + 1
F5 (x)	1 x 2 +   3 x + 1
F6 (x)	3 x 2 +   4 x + 1
F7 (x)	1 x 3 +   6 x 2 +   5 x + 1
F8 (x)	4 x 3 + 10 x 2 +   6 x + 1
F9 (x)	1 x 4 + 10 x 3 + 15 x 2 +   7 x + 1
F10 (x)	5 x 4 + 20 x 3 + 21 x 2 +   8 x + 1
F11 (x)	1 x 5 + 15 x 4 + 35 x 3 + 28 x 2 +   9 x + 1
F12 (x)	6 x 5 + 35 x 4 + 56 x 3 + 36 x 2 + 10 x + 1

The sequence of Fibonacci polynomials 

Fn (x)

is a sequence of polynomials defined by the recurrence relation

${\begin{array}{l}\displaystyle {F_{n}(x):={\begin{cases}0,&{\mbox{if }}n=0,\\1,&{\mbox{if }}n=1,\\F_{n-1}(x)+x\,F_{n-2}(x),&{\mbox{if }}n\geq 2,\end{cases}}}\end{array}}$

where

• the degree of 

  Fn (x), n   ≥   1,

  is 

  ⌈n / 2⌉  −  1

  ;
•  

  Fn = Fn (1), n   ≥   0,

  where 

  Fn

  is the 

  n

  th Fibonacci number.
  
  

If you look at the Fibonacci polynomials triangle, you will see that the rising diagonals corresponding to odd 

n

are the "(1, 1)-Pascal polynomials". And the column of degree 

1

have natural numbers as coefficients, the column of degree 

2

have triangular numbers as coefficients, the column of degree 

3

have tetrahedral numbers as coefficients, and so on... (Cf. rows of (1, 1)-Pascal triangle, i.e. Pascal triangle.)

This yields the infinite sequence of finite sequences

 

{{0}, {1}, {1}, {1, 1}, {2, 1}, {1, 3, 1}, {3, 4, 1}, {1, 6, 5, 1}, {4, 10, 6, 1}, {1, 10, 15, 7, 1}, {5, 20, 21, 8, 1}, {1, 15, 35, 28, 9, 1}, {6, 35, 56, 36, 10, 1}, ... },

whose concatenation yields the infinite sequence (see A102426)

 

{ 0, 1, 1, 1, 1, 2, 1, 1, 3, 1, 3, 4, 1, 1, 6, 5, 1, 4, 10, 6, 1, 1, 10, 15, 7, 1, 5, 20, 21, 8, 1, 1, 15, 35, 28, 9, 1, 6, 35, 56, 36, 10, 1, ... }.

Fibonacci rabbits per generation

If we interpret the Fibonacci numbers 

Fn

as the number of [male-female] pairs of Fibonacci rabbits at the beginning of the 

n

th month, where each pair that is at least 2 months old begets a [male-female] pair of newborn Fibonacci rabbits, then the coefficient 

ak

of the term 

ak x k

of the Fibonacci polynomial 

Fn (x)

indicates the number of [male-female] pairs of Fibonacci rabbits which belong to the 

k

th generation.

Tree of newborn (red) and mature (green) "[male-female] pairs of Fibonacci rabbits" (node labels show the generation   k ) 0 0 0 1 0 1 1 0 1 1 1 2 0 1 1 1 2 1 2 2

For example

$F_{6}(x)=3x^{2}+4x+1\,$

means that at the beginning of the 6^th month, we have

• 3 pairs belonging to the 2^nd generation (offsprings of the offsprings of the original pair),
• 4 pairs belonging to the 1^st generation (offsprings of the original pair),
• 1 pair belonging to the 0^th generation (the original pair),

for a total of 8 [male-female] pairs of Fibonacci rabbits.

At the beginning of the 

n

th month, we have 

⌈n / 2⌉

generations, i.e. generations 

0

to 

⌈n / 2⌉  −  1

, of [male-female] pairs of Fibonacci rabbits. The original pair, i.e. the 0^th generation, was a [male-female] pair of newborn Fibonacci rabbits, i.e. 0 months old, which was dropped on the island at the beginning of the first month. (At the beginning of the 0^th month, there were no rabbits.)

Reading the rows of generation labels (the labels of row 

n

are the labels of row 

n  −  1

appended with the incremented labels of row 

n  −  2

) left-to-right yields the infinite sequence (see A??????)

 

{0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 2, 1, 2, 2, ...}.

Reading the rows of generation labels right-to-left yields the infinite sequence (see A200650 

(n  +  1), n   ≥   1

)

 

{0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 0, 2, 2, 1, 2, 1, 1, 1, 0, ...}.

Formulae

${\begin{array}{l}\displaystyle {F_{n}(x)=\sum _{k=0}^{\lceil {\frac {n}{2}}\rceil -1}T_{(1,1)}\left({{\bigg \lfloor }{\frac {n}{2}}{\bigg \rfloor }+k},{{\bigg \lfloor }{\frac {n-1}{2}}{\bigg \rfloor }-k}\right)\,x^{2k+1-(n{\bmod {2}})}=\sum _{k=0}^{\lceil {\frac {n}{2}}\rceil -1}{\binom {\lfloor {\frac {n}{2}}\rfloor +k}{\lfloor {\frac {n-1}{2}}\rfloor -k}}\,x^{2k+1-(n{\bmod {2}})},\quad n\geq 1,}\end{array}}$

where 

T(1, 1) (n, k) = (  nk  )

is the term from column 

k

of row 

n

of the (1, 1)-Pascal triangle (i.e. Pascal triangle), and 

(  nk  )

is a binomial coefficient.

Generating function

The ordinary generating function for the sequence of Fibonacci polynomials is

${\begin{array}{l}\displaystyle {G_{\{F_{n}(x)\}}(t)=\sum _{n=0}^{\infty }F_{n}(x)\,t^{n}={\frac {t}{1-t\,(x+t)}}={\frac {t}{1-(t^{1}x^{1}+t^{2}x^{0})}}.}\end{array}}$

We may also observe the following relation

${\begin{array}{l}\displaystyle {{\frac {t}{1-t\,(x+t)}}=\sum _{n=0}^{\infty }{\left({\frac {t}{1-t^{2}}}\right)}^{n+1}x^{n}=\sum _{n=0}^{\infty }{\left(\sum _{k=0}^{\infty }t^{2k+1}\right)}^{n+1}x^{n},}\end{array}}$

where

${\begin{array}{l}\displaystyle {\sum _{k=0}^{\infty }t^{2k+1}={\frac {t}{1-t^{2}}}.}\end{array}}$

See also

• (1, 1)-Pascal polynomials (compared to Fibonacci polynomials)
• (1, 2)-Pascal polynomials (compared to Lucas polynomials)

• A049310 Triangle of coefficients of Chebyshev polynomials 

  S (n, x) := U (n, x / 2)

  (exponents in increasing order). Unsigned triangle 

  | a (n, m) |

  has Fibonacci polynomials (definition according to MathWorld) 

  F (n  +  1, x)

  as row polynomials.

• Pascal's triangle

• Polynomial sequences (number sequences described by a polynomial formula (not to be confused with sequences of polynomials)