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# Fibonacci polynomials

[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.

## Contents

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 1st 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

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