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

Dickson polynomials

Dickson polynomials of the first kind

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

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

The first few Dickson polynomials of the first kind are

$D_{0}(x,\alpha )=2\,$

$D_{1}(x,\alpha )=x\,$

$D_{2}(x,\alpha )=x^{2}-2\alpha \,$

$D_{3}(x,\alpha )=x^{3}-3x\alpha \,$

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

Dickson polynomials of the first kind (with 𝞪 = 1)

The Dickson polynomials (with 𝞪 = 1) $\scriptstyle D_{n}(x)\,\equiv \,D_{n}(x,1)\,$ are given by

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

The first few Dickson polynomials (with 𝞪 = 1) are

$D_{0}(x)=2\,$

$D_{1}(x)=x\,$

$D_{2}(x)=x^{2}-2\,$

$D_{3}(x)=x^{3}-3x\,$

$D_{4}(x)=x^{4}-4x^{2}+2\,$

Dickson polynomials (with 𝞪 = 1), $\scriptstyle D_{n}(x)\,$, are equivalent to Chebyshev polynomials $\scriptstyle T_{n}(x)\,$, with a slight and trivial change of variable

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

Dickson polynomials of the second kind

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

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

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

$E_{0}(x,\alpha )=1\,$

$E_{1}(x,\alpha )=x\,$

$E_{2}(x,\alpha )=x^{2}-\alpha \,$

$E_{3}(x,\alpha )=x^{3}-2x\alpha \,$

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

Fibonacci polynomials

Main article page: Fibonacci polynomials

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

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

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

The ordinary generating function for the sequence of Fibonacci polynomials is

$G_{\{F_{n}(x)\}}(t)=\sum _{n=0}^{\infty }F_{n}(x)~t^{n}={\frac {t}{1-t(x+t)}}.\,$

Fibonacci polynomials

$n\,$	$F_{n}(x)=\,$	$a_{0}x^{0}\,$	$+a_{1}x^{1}\,$	$+a_{2}x^{2}\,$	$+a_{3}x^{3}\,$	$+a_{4}x^{4}\,$	$+a_{5}x^{5}\,$	$+a_{6}x^{6}\,$	$+a_{7}x^{7}\,$	$+a_{8}x^{8}\,$	$+a_{9}x^{9}\,$	$+a_{10}x^{10}\,$	$+a_{11}x^{11}\,$	$+a_{12}x^{12}\,$
0	$F_{0}(x)=\,$	$+0\,$
1	$F_{1}(x)=\,$	$+1\,$	$+0\,$
2	$F_{2}(x)=\,$		$+x\,$	$+0\,$
3	$F_{3}(x)=\,$	$+1\,$		$+x^{2}\,$	$+0\,$
4	$F_{4}(x)=\,$		$+2x\,$		$+x^{3}\,$	$+0\,$
5	$F_{5}(x)=\,$	$+1\,$		$+3x^{2}\,$		$+x^{4}\,$	$+0\,$
6	$F_{6}(x)=\,$		$+3x\,$		$+4x^{3}\,$		$+x^{5}\,$	$+0\,$
7	$F_{7}(x)=\,$	$+1\,$		$+6x^{2}\,$		$+5x^{4}\,$		$+x^{6}\,$	$+0\,$
8	$F_{8}(x)=\,$		$+4x\,$		$+10x^{3}\,$		$+6x^{5}\,$		$+x^{7}\,$	$+0\,$
9	$F_{9}(x)=\,$	$+1\,$		$+10x^{2}\,$		$+15x^{4}\,$		$+7x^{6}\,$		$+x^{8}\,$	$+0\,$
10	$F_{10}(x)=\,$		$+5x\,$		$+20x^{3}\,$		$+21x^{5}\,$		$+8x^{7}\,$		$+x^{9}\,$	$+0\,$
11	$F_{11}(x)=\,$	$+1\,$		$+15x^{2}\,$		$+35x^{4}\,$		$+28x^{6}\,$		$+9x^{8}\,$		$+x^{10}\,$	$+0\,$
12	$F_{12}(x)=\,$		$+6x\,$		$+35x^{3}\,$		$+56x^{5}\,$		$+36x^{7}\,$		$+10x^{9}\,$		$+x^{11}\,$	$+0\,$

If you look at the Fibonacci polynomials triangle, you will see that the rising diagonals corresponding to odd $\scriptstyle n\,$ 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 Édouard Lucas in 1878 to study linear recurrence relations, prime numbers, and other aspects of mathematics.

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

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

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

The ordinary generating function of the Lucas polynomials is

$G_{\{L_{n}(x)\}}(t)=\sum _{n=0}^{\infty }L_{n}(x)~t^{n}={\frac {2-xt}{1-t(x+t)}}.\,$

Lucas polynomials

$n\,$	$L_{n}(x)=\,$	$a_{0}x^{0}\,$	$+a_{1}x^{1}\,$	$+a_{2}x^{2}\,$	$+a_{3}x^{3}\,$	$+a_{4}x^{4}\,$	$+a_{5}x^{5}\,$	$+a_{6}x^{6}\,$	$+a_{7}x^{7}\,$	$+a_{8}x^{8}\,$	$+a_{9}x^{9}\,$	$+a_{10}x^{10}\,$	$+a_{11}x^{11}\,$	$+a_{12}x^{12}\,$
0	$L_{0}(x)=\,$	$+2\,$
1	$L_{1}(x)=\,$		$+x\,$
2	$L_{2}(x)=\,$	$+2\,$		$+x^{2}\,$
3	$L_{3}(x)=\,$		$+3x\,$		$+x^{3}\,$
4	$L_{4}(x)=\,$	$+2\,$		$+4x^{2}\,$		$+x^{4}\,$
5	$L_{5}(x)=\,$		$+5x\,$		$+5x^{3}\,$		$+x^{5}\,$
6	$L_{6}(x)=\,$	$+2\,$		$+9x^{2}\,$		$+6x^{4}\,$		$+x^{6}\,$
7	$L_{7}(x)=\,$		$+7x\,$		$+14x^{3}\,$		$+7x^{5}\,$		$+x^{7}\,$
8	$L_{8}(x)=\,$	$+2\,$		$+16x^{2}\,$		$+20x^{4}\,$		$+8x^{6}\,$		$+x^{8}\,$
9	$L_{9}(x)=\,$		$+9x\,$		$+30x^{3}\,$		$+27x^{5}\,$		$+9x^{7}\,$		$+x^{9}\,$
10	$L_{10}(x)=\,$	$+2\,$		$+25x^{2}\,$		$+50x^{4}\,$		$+35x^{6}\,$		$+10x^{8}\,$		$+x^{10}\,$
11	$L_{11}(x)=\,$		$+11x\,$		$+55x^{3}\,$		$+77x^{5}\,$		$+44x^{7}\,$		$+11x^{9}\,$		$+x^{11}\,$
12	$L_{12}(x)=\,$	$+2\,$		$+36x^{2}\,$		$+105x^{4}\,$		$+112x^{6}\,$		$+54x^{8}\,$		$+12x^{10}\,$		$+x^{12}\,$

If you look at the Lucas polynomials triangle, you will see that the rising diagonals corresponding to even $\scriptstyle n\,$ 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

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

The Boubaker polynomials can be defined by the recurrence relation

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

They are given by the closed form formula

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

The ordinary generating function is

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

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

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

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

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