The Chebyshev polynomials have many special cases which have been studied under other names, most famously the Lucas polynomials.
Denote by
the Chebyshev polynomials of the first kind and by
the Chebyshev polynomials of the second kind.
Dickson polynomials
Dickson polynomials of the first kind
Dickson polynomials (of the first kind)
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}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/aa39eae407c7fe1150285097b15d7166b729615b)
The first few Dickson polynomials of the first kind are
![{\displaystyle D_{0}(x,\alpha )=2\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/6c09f070f602c3fd2dc5e3b4e220568c62d4f5b8)
![{\displaystyle D_{1}(x,\alpha )=x\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ee7d3270c0acb120220c75a21349708eaf4a405f)
![{\displaystyle D_{2}(x,\alpha )=x^{2}-2\alpha \,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/65ef7f03733e3b16f12b4e7c64951c7a4422d093)
![{\displaystyle D_{3}(x,\alpha )=x^{3}-3x\alpha \,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/513e339693d803b4866b3dc6c6771550c7c1ab4a)
![{\displaystyle D_{4}(x,\alpha )=x^{4}-4x^{2}\alpha +2\alpha ^{2}.\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/404fa9dc9b52553a9e99806e60ad8f298b9e8a5b)
Dickson polynomials of the first kind (with πͺ = 1)
The Dickson polynomials (with πͺ = 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}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b9affbc2e5724d79acd818aa881ad645aad33247)
The first few Dickson polynomials (with πͺ = 1) are
![{\displaystyle D_{0}(x)=2\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f16164cc3a7676396b09c7a76f6c912e88cee366)
![{\displaystyle D_{1}(x)=x\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8663fa7b1fbca42e16b6a31522b77f31db55fe37)
![{\displaystyle D_{2}(x)=x^{2}-2\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/171f041f20ad37d9f39674928ae4b0c0553af019)
![{\displaystyle D_{3}(x)=x^{3}-3x\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/dc8e03b8548449c281160e874daf15f6f1187f0e)
![{\displaystyle D_{4}(x)=x^{4}-4x^{2}+2\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/86968fac47f7db9f2ba3ca0480cba7219d9dcbe8)
Dickson polynomials (with πͺ = 1),[1]
[2]
[3]
[4]
, are equivalent to Chebyshev polynomials
, with a slight and trivial change of variable
![{\displaystyle D_{n}(2x)=2T_{n}(x)\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8b8043948297007048193a8aba487868e8875fdb)
Dickson polynomials of the second kind
Dickson polynomials of the second kind
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.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/3ad76348225edd116e91409b3c8eecc86d76cf70)
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\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/45e57d870e878a5aa72b02ad00ef02ac60805191)
![{\displaystyle E_{1}(x,\alpha )=x\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/672625ce09bcec96db5ebde2c135c221e81bfe1d)
![{\displaystyle E_{2}(x,\alpha )=x^{2}-\alpha \,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b5b5ef4a89ef4d9a3ac85497c5915a0ffe7abd17)
![{\displaystyle E_{3}(x,\alpha )=x^{3}-2x\alpha \,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/2562bb49966c02cd028cfbad8e98db8d464f94e1)
![{\displaystyle E_{4}(x,\alpha )=x^{4}-3x^{2}\alpha +\alpha ^{2}.\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b123a8f5e5198a79ad875969919250963f6dfa74)
Fibonacci polynomials
- Main article page: Fibonacci polynomials
The sequence of Fibonacci polynomials
[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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/406ac4742b2fba12462b78f5ee404b6ae65c218b)
![{\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}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a733e2e0ce0f30742a0f415b34d9de52f1788264)
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)}}.\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a047b224a11d607ae0dfd9eac04daa4387a325f7)
Fibonacci polynomials
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If you look at the Fibonacci polynomials triangle, you will see that the rising diagonals corresponding to odd
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 Γdouard Lucas in 1878 to study linear recurrence relations, prime numbers, and other aspects of mathematics.
The sequence of Lucas polynomials
[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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/91a490b99b677ad24bfe780d5d41f8dfea7bf52d)
![{\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}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/1776492715ad6096a8c7a93e73f44d3bc9ca4256)
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)}}.\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/345facadda954e2a7f91f64f8749864e7fe71eb4)
Lucas polynomials
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If you look at the Lucas polynomials triangle, you will see that the rising diagonals corresponding to even
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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/edefbd3f1c5db32dd63090f2fb5fad1e7e3e0ba2)
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}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/bc418eebb0d8f9f9af011d31775616e57951a43c)
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}.\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/76419420dc5034021c689e6506d1ed249cdc2574)
The ordinary generating function is
![{\displaystyle \sum _{n=0}^{\infty }B_{n}(x)~t^{n}={\frac {1+3t^{2}}{1-t(t-x)}}.\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d26fa7824f8111fe634e763105a7fc70afdbc130)
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.\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/530340e79072b721541ebc664e6531b960659b72)
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).\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c969e99f5613ce400c8d5b3cd653cf6c5d5aa292)
Notes
- β [1]
- β [2]
- β [3]
- β [4]
- β Eric W. Weisstein, Fibonacci Polynomial, from MathWorld — A Wolfram Web Resource.
- β Eric W. Weisstein, Lucas Polynomial, from MathWorld — A Wolfram Web Resource.
- 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