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# Special cases of the Chebyshev polynomials

The Chebyshev polynomials have many special cases which have been studied under other names, most famously the Lucas polynomials.

Denote by ${\displaystyle \scriptstyle T_{n}(x)\,}$ the Chebyshev polynomials of the first kind and by ${\displaystyle \scriptstyle U_{n}(x)\,}$ the Chebyshev polynomials of the second kind.

## Dickson polynomials

### Dickson polynomials of the first kind

Dickson polynomials (of the first kind) ${\displaystyle \scriptstyle D_{n}(x,\,\alpha )\,}$ are defined by

${\displaystyle D_{n}(x,\alpha )={\begin{cases}2,&{\mbox{if }}n=0,\\\sum _{p=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {n}{n-p}}~{\binom {n-p}{p}}~(-\alpha )^{p}~x^{n-2p},&{\mbox{if }}n\geq 1.\end{cases}}\,}$

The first few Dickson polynomials of the first kind are

${\displaystyle D_{0}(x,\alpha )=2\,}$
${\displaystyle D_{1}(x,\alpha )=x\,}$
${\displaystyle D_{2}(x,\alpha )=x^{2}-2\alpha \,}$
${\displaystyle D_{3}(x,\alpha )=x^{3}-3x\alpha \,}$
${\displaystyle D_{4}(x,\alpha )=x^{4}-4x^{2}\alpha +2\alpha ^{2}.\,}$

#### Dickson polynomials of the first kind (with πͺ = 1)

The Dickson polynomials (with πͺ = 1) ${\displaystyle \scriptstyle D_{n}(x)\,\equiv \,D_{n}(x,1)\,}$ are given by

${\displaystyle D_{n}(x)\equiv D_{n}(x,1)={\begin{cases}2,&{\mbox{if }}n=0,\\\sum _{p=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {n}{n-p}}~{\binom {n-p}{p}}~(-1)^{p}~x^{n-2p},&{\mbox{if }}n\geq 1.\end{cases}}\,}$

The first few Dickson polynomials (with πͺ = 1) are

${\displaystyle D_{0}(x)=2\,}$
${\displaystyle D_{1}(x)=x\,}$
${\displaystyle D_{2}(x)=x^{2}-2\,}$
${\displaystyle D_{3}(x)=x^{3}-3x\,}$
${\displaystyle D_{4}(x)=x^{4}-4x^{2}+2\,}$

Dickson polynomials (with πͺ = 1),[1] [2] [3] [4] ${\displaystyle \scriptstyle D_{n}(x)\,}$, are equivalent to Chebyshev polynomials ${\displaystyle \scriptstyle T_{n}(x)\,}$, with a slight and trivial change of variable

${\displaystyle D_{n}(2x)=2T_{n}(x)\,}$

### Dickson polynomials of the second kind

Dickson polynomials of the second kind ${\displaystyle \scriptstyle E_{n}(x,\,\alpha )\,}$ are defined by

${\displaystyle E_{n}(x,\alpha )=\sum _{p=0}^{\lfloor {\frac {n}{2}}\rfloor }{\binom {n-p}{p}}~(-\alpha )^{p}~x^{n-2p},\quad n\geq 0.}$

They have not been studied much, and have properties similar to those of Dickson polynomials of the first kind.

The first few Dickson polynomials of the second kind are

${\displaystyle E_{0}(x,\alpha )=1\,}$
${\displaystyle E_{1}(x,\alpha )=x\,}$
${\displaystyle E_{2}(x,\alpha )=x^{2}-\alpha \,}$
${\displaystyle E_{3}(x,\alpha )=x^{3}-2x\alpha \,}$
${\displaystyle E_{4}(x,\alpha )=x^{4}-3x^{2}\alpha +\alpha ^{2}.\,}$

## Fibonacci polynomials

Main article page: Fibonacci polynomials

The sequence of Fibonacci polynomials ${\displaystyle \scriptstyle F_{n}(x)\,}$[5] is a sequence of polynomials defined by the recurrence relation (compare with the Chebyshev polynomials of the second kind, at the right)

${\displaystyle U_{n}(x)\equiv {\begin{cases}1x^{0}=1,&{\mbox{if }}n=0,\\2x^{1}=2x,&{\mbox{if }}n=1,\\2x^{1}~U_{n-1}(x)-1x^{0}~U_{n-2}(x),&{\mbox{if }}n\geq 2.\end{cases}}}$
${\displaystyle F_{n}(x)={\begin{cases}0x^{0}=0,&{\mbox{if }}n=0,\\1x^{0}+0x^{1}=1,&{\mbox{if }}n=1,\\x^{1}F_{n-1}(x)+x^{0}F_{n-2}(x),&{\mbox{if }}n\geq 2.\end{cases}}\,}$

The ordinary generating function for the sequence of Fibonacci polynomials is

${\displaystyle G_{\{F_{n}(x)\}}(t)=\sum _{n=0}^{\infty }F_{n}(x)~t^{n}={\frac {t}{1-t(x+t)}}.\,}$
Fibonacci polynomials
${\displaystyle n\,}$ ${\displaystyle F_{n}(x)=\,}$ ${\displaystyle a_{0}x^{0}\,}$ ${\displaystyle +a_{1}x^{1}\,}$ ${\displaystyle +a_{2}x^{2}\,}$ ${\displaystyle +a_{3}x^{3}\,}$ ${\displaystyle +a_{4}x^{4}\,}$ ${\displaystyle +a_{5}x^{5}\,}$ ${\displaystyle +a_{6}x^{6}\,}$ ${\displaystyle +a_{7}x^{7}\,}$ ${\displaystyle +a_{8}x^{8}\,}$ ${\displaystyle +a_{9}x^{9}\,}$ ${\displaystyle +a_{10}x^{10}\,}$ ${\displaystyle +a_{11}x^{11}\,}$ ${\displaystyle +a_{12}x^{12}\,}$
0 ${\displaystyle F_{0}(x)=\,}$ ${\displaystyle +0\,}$
1 ${\displaystyle F_{1}(x)=\,}$ ${\displaystyle +1\,}$ ${\displaystyle +0\,}$
2 ${\displaystyle F_{2}(x)=\,}$   ${\displaystyle +x\,}$ ${\displaystyle +0\,}$
3 ${\displaystyle F_{3}(x)=\,}$ ${\displaystyle +1\,}$   ${\displaystyle +x^{2}\,}$ ${\displaystyle +0\,}$
4 ${\displaystyle F_{4}(x)=\,}$   ${\displaystyle +2x\,}$   ${\displaystyle +x^{3}\,}$ ${\displaystyle +0\,}$
5 ${\displaystyle F_{5}(x)=\,}$ ${\displaystyle +1\,}$   ${\displaystyle +3x^{2}\,}$   ${\displaystyle +x^{4}\,}$ ${\displaystyle +0\,}$
6 ${\displaystyle F_{6}(x)=\,}$   ${\displaystyle +3x\,}$   ${\displaystyle +4x^{3}\,}$   ${\displaystyle +x^{5}\,}$ ${\displaystyle +0\,}$
7 ${\displaystyle F_{7}(x)=\,}$ ${\displaystyle +1\,}$   ${\displaystyle +6x^{2}\,}$   ${\displaystyle +5x^{4}\,}$   ${\displaystyle +x^{6}\,}$ ${\displaystyle +0\,}$
8 ${\displaystyle F_{8}(x)=\,}$   ${\displaystyle +4x\,}$   ${\displaystyle +10x^{3}\,}$   ${\displaystyle +6x^{5}\,}$   ${\displaystyle +x^{7}\,}$ ${\displaystyle +0\,}$
9 ${\displaystyle F_{9}(x)=\,}$ ${\displaystyle +1\,}$   ${\displaystyle +10x^{2}\,}$   ${\displaystyle +15x^{4}\,}$   ${\displaystyle +7x^{6}\,}$   ${\displaystyle +x^{8}\,}$ ${\displaystyle +0\,}$
10 ${\displaystyle F_{10}(x)=\,}$   ${\displaystyle +5x\,}$   ${\displaystyle +20x^{3}\,}$   ${\displaystyle +21x^{5}\,}$   ${\displaystyle +8x^{7}\,}$   ${\displaystyle +x^{9}\,}$ ${\displaystyle +0\,}$
11 ${\displaystyle F_{11}(x)=\,}$ ${\displaystyle +1\,}$   ${\displaystyle +15x^{2}\,}$   ${\displaystyle +35x^{4}\,}$   ${\displaystyle +28x^{6}\,}$   ${\displaystyle +9x^{8}\,}$   ${\displaystyle +x^{10}\,}$ ${\displaystyle +0\,}$
12 ${\displaystyle F_{12}(x)=\,}$   ${\displaystyle +6x\,}$   ${\displaystyle +35x^{3}\,}$   ${\displaystyle +56x^{5}\,}$   ${\displaystyle +36x^{7}\,}$   ${\displaystyle +10x^{9}\,}$   ${\displaystyle +x^{11}\,}$ ${\displaystyle +0\,}$

If you look at the Fibonacci polynomials triangle, you will see that the rising diagonals corresponding to odd ${\displaystyle \scriptstyle n\,}$ 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.)

## Lucas polynomials

Main article page: Lucas polynomials

The Lucas polynomials (or Cardan polynomials) were created by Γdouard Lucas in 1878 to study linear recurrence relations, prime numbers, and other aspects of mathematics.

The sequence of Lucas polynomials ${\displaystyle \scriptstyle L_{n}(x)\,}$[6] is a sequence of polynomials defined by the recurrence relation (compare with Chebyshev polynomials of the first kind recurrence on the right)

${\displaystyle T_{n}(x)\equiv {\begin{cases}1x^{0}=1,&{\mbox{if }}n=0,\\1x^{1}=x,&{\mbox{if }}n=1,\\2x^{1}~T_{n-1}(x)-1x^{0}~T_{n-2}(x),&{\mbox{if }}n\geq 2.\end{cases}}}$
${\displaystyle L_{n}(x)={\begin{cases}2x^{0}=2,&{\mbox{if }}n=0,\\1x^{1}=x,&{\mbox{if }}n=1,\\x^{1}L_{n-1}(x)+x^{0}L_{n-2}(x),&{\mbox{if }}n\geq 2.\end{cases}}\,}$

The ordinary generating function of the Lucas polynomials is

${\displaystyle G_{\{L_{n}(x)\}}(t)=\sum _{n=0}^{\infty }L_{n}(x)~t^{n}={\frac {2-xt}{1-t(x+t)}}.\,}$
Lucas polynomials
${\displaystyle n\,}$ ${\displaystyle L_{n}(x)=\,}$ ${\displaystyle a_{0}x^{0}\,}$ ${\displaystyle +a_{1}x^{1}\,}$ ${\displaystyle +a_{2}x^{2}\,}$ ${\displaystyle +a_{3}x^{3}\,}$ ${\displaystyle +a_{4}x^{4}\,}$ ${\displaystyle +a_{5}x^{5}\,}$ ${\displaystyle +a_{6}x^{6}\,}$ ${\displaystyle +a_{7}x^{7}\,}$ ${\displaystyle +a_{8}x^{8}\,}$ ${\displaystyle +a_{9}x^{9}\,}$ ${\displaystyle +a_{10}x^{10}\,}$ ${\displaystyle +a_{11}x^{11}\,}$ ${\displaystyle +a_{12}x^{12}\,}$
0 ${\displaystyle L_{0}(x)=\,}$ ${\displaystyle +2\,}$
1 ${\displaystyle L_{1}(x)=\,}$   ${\displaystyle +x\,}$
2 ${\displaystyle L_{2}(x)=\,}$ ${\displaystyle +2\,}$   ${\displaystyle +x^{2}\,}$
3 ${\displaystyle L_{3}(x)=\,}$   ${\displaystyle +3x\,}$   ${\displaystyle +x^{3}\,}$
4 ${\displaystyle L_{4}(x)=\,}$ ${\displaystyle +2\,}$   ${\displaystyle +4x^{2}\,}$   ${\displaystyle +x^{4}\,}$
5 ${\displaystyle L_{5}(x)=\,}$   ${\displaystyle +5x\,}$   ${\displaystyle +5x^{3}\,}$   ${\displaystyle +x^{5}\,}$
6 ${\displaystyle L_{6}(x)=\,}$ ${\displaystyle +2\,}$   ${\displaystyle +9x^{2}\,}$   ${\displaystyle +6x^{4}\,}$   ${\displaystyle +x^{6}\,}$
7 ${\displaystyle L_{7}(x)=\,}$   ${\displaystyle +7x\,}$   ${\displaystyle +14x^{3}\,}$   ${\displaystyle +7x^{5}\,}$   ${\displaystyle +x^{7}\,}$
8 ${\displaystyle L_{8}(x)=\,}$ ${\displaystyle +2\,}$   ${\displaystyle +16x^{2}\,}$   ${\displaystyle +20x^{4}\,}$   ${\displaystyle +8x^{6}\,}$   ${\displaystyle +x^{8}\,}$
9 ${\displaystyle L_{9}(x)=\,}$   ${\displaystyle +9x\,}$   ${\displaystyle +30x^{3}\,}$   ${\displaystyle +27x^{5}\,}$   ${\displaystyle +9x^{7}\,}$   ${\displaystyle +x^{9}\,}$
10 ${\displaystyle L_{10}(x)=\,}$ ${\displaystyle +2\,}$   ${\displaystyle +25x^{2}\,}$   ${\displaystyle +50x^{4}\,}$   ${\displaystyle +35x^{6}\,}$   ${\displaystyle +10x^{8}\,}$   ${\displaystyle +x^{10}\,}$
11 ${\displaystyle L_{11}(x)=\,}$   ${\displaystyle +11x\,}$   ${\displaystyle +55x^{3}\,}$   ${\displaystyle +77x^{5}\,}$   ${\displaystyle +44x^{7}\,}$   ${\displaystyle +11x^{9}\,}$   ${\displaystyle +x^{11}\,}$
12 ${\displaystyle L_{12}(x)=\,}$ ${\displaystyle +2\,}$   ${\displaystyle +36x^{2}\,}$   ${\displaystyle +105x^{4}\,}$   ${\displaystyle +112x^{6}\,}$   ${\displaystyle +54x^{8}\,}$   ${\displaystyle +12x^{10}\,}$   ${\displaystyle +x^{12}\,}$

If you look at the Lucas polynomials triangle, you will see that the rising diagonals corresponding to even ${\displaystyle \scriptstyle n\,}$ are the "(1,2)-Pascal polynomials". And the column of degree 1 have odd numbers as coefficients, the column of degree 2 have square numbers as coefficients, the column of degree 3 have square pyramidal numbers as coefficients, and so on... (Cf. rows of (1,2)-Pascal triangle, i.e. Lucas triangle.)

## Boubaker polynomials

{\displaystyle {\begin{aligned}B_{0}(x)&=1\\B_{1}(x)&=x\\B_{2}(x)&=x^{2}+2\\B_{3}(x)&=x^{3}+x\\B_{4}(x)&=x^{4}-2\\B_{5}(x)&=x^{5}-x^{3}-3x\\\end{aligned}}}

The Boubaker polynomials can be defined by the recurrence relation[7]

${\displaystyle B_{n}(x)={\begin{cases}1,&{\mbox{if }}n=0,\\x,&{\mbox{if }}n=1,\\x^{2}+2,&{\mbox{if }}n=2,\\xB_{n-1}(x)-B_{n-2}(x),&{\mbox{if }}n\geq 3.\end{cases}}\,}$

They are given by the closed form formula

${\displaystyle B_{n}(x)=\sum _{p=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {n-4p}{n-p}}{\binom {n-p}{p}}~(-1)^{p}~x^{n-2p}.\,}$
${\displaystyle \sum _{n=0}^{\infty }B_{n}(x)~t^{n}={\frac {1+3t^{2}}{1-t(t-x)}}.\,}$

In terms of the Chebyshev polynomials of the second kind, we have

${\displaystyle B_{n}(x)=U_{n}({\tfrac {x}{2}})+3~U_{n-2}({\tfrac {x}{2}}),\quad n\geq 2.\,}$

In terms of both the Chebyshev polynomials of the first kind and the Chebyshev polynomials of the second kind, we have

${\displaystyle B_{n}(2x)=-2~T_{n}(x)+4x~U_{n-1}(x).\,}$

## Notes

1. β [1]
2. β
3. β
4. β
5. β Eric W. Weisstein, Fibonacci Polynomial, from MathWorld — A Wolfram Web Resource.
6. β Eric W. Weisstein, Lucas Polynomial, from MathWorld — A Wolfram Web Resource.
7. β O.D. Oyodum, O.B. Awojoyogbe, M.K. Dada, J.N. Magnuson, Comment on "Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition" by K. Boubaker, A. Chaouachi, M. Amlouk and H. Bouzouita. On the earliest definition of the Boubaker polynomials, Eur. Phys. J. Appl. Phys., Volume 46, DOI:10.1051/epjap/2009036