Chebyshev polynomials

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The Chebyshev polynomials, named after Pafnuty Chebyshev, are sequences of polynomials (of orthogonal polynomials) which are related to de Moivre's formula and which are easily defined recursively, like Fibonacci or Lucas numbers. One usually distinguishes between Chebyshev polynomials of the first kind, which are denoted $\scriptstyle T_{n}(x)\,$, and Chebyshev polynomials of the second kind, which are denoted $\scriptstyle U_{n}(x)\,$. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebychev (French) or Tschebyschow (German.)

Chebyshev polynomials of the first kind

The Chebyshev polynomials of the first kind are defined by the recurrence relation

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

The first few Chebyshev polynomials of the first kind are

$T_{0}(x)=1\,$

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

$T_{2}(x)=2x^{2}-1\,$

$T_{3}(x)=4x^{3}-3x\,$

$T_{4}(x)=8x^{4}-8x^{2}+1\,$

$T_{5}(x)=16x^{5}-20x^{3}+5x\,$

$T_{6}(x)=32x^{6}-48x^{4}+18x^{2}-1\,$

$T_{7}(x)=64x^{7}-112x^{5}+56x^{3}-7x\,$

$T_{8}(x)=128x^{8}-256x^{6}+160x^{4}-32x^{2}+1\,$

$T_{9}(x)=256x^{9}-576x^{7}+432x^{5}-120x^{3}+9x\,$

A closed-form formula (would be a Binet formula of the first type, but for the factor $\scriptstyle x\,$) (Cf. Lucas numbers#Binet's closed-form formula) giving the Chebyshev polynomials of the first kind is

$T_{n}(x)=x~\left\{{\frac {(\rho _{+})^{n}+(\rho _{-})^{n}}{(\rho _{+})+(\rho _{-})}}\right\}=x~\left\{{\frac {(x+{\sqrt {x^{2}-1}})^{n}+(x-{\sqrt {x^{2}-1}})^{n}}{(x+{\sqrt {x^{2}-1}})+(x-{\sqrt {x^{2}-1}})}}\right\}=x~\left\{{\frac {(x+{\sqrt {x^{2}-1}})^{n}+(x-{\sqrt {x^{2}-1}})^{n}}{2x}}\right\},\,$

where $\scriptstyle \rho _{+}\,$ and $\scriptstyle \rho _{-}\,$ are the roots of the quadratic polynomial in terms of $\scriptstyle t\,$

$1-2tx+t^{2}=0\,$

The ordinary generating function for $\scriptstyle T_{n}(x)\,$ is

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

The exponential generating function for $\scriptstyle T_{n}(x)\,$ is

$E_{\{T_{n}(x)\}}(t)=\sum _{n=0}^{\infty }T_{n}(x)~{\frac {t^{n}}{n!}}={\frac {1}{2}}\left(e^{(\rho _{+})t}+e^{(\rho _{-})t}\right)={\frac {1}{2}}\left(e^{(x+{\sqrt {x^{2}-1}})t}+e^{(x-{\sqrt {x^{2}-1}})t}\right)=e^{x}~\cosh({\sqrt {x^{2}-1}}~t).\,$

Triangle of coefficients of Chebyshev polynomials of the first kind

Chebyshev polynomials of the first kind

$n\,$	$T_{n}(x)=\,$	$a_{0}x^{0}\,$	$+a_{1}x^{1}\,$	$+a_{2}x^{2}\,$	$+a_{3}x^{3}\,$	$+a_{4}x^{4}\,$	$+a_{5}x^{5}\,$	$+a_{6}x^{6}\,$	$+a_{7}x^{7}\,$	$+a_{8}x^{8}\,$	$+a_{9}x^{9}\,$	$+a_{10}x^{10}\,$	$+a_{11}x^{11}\,$	$+a_{12}x^{12}\,$
0	$T_{0}(x)=\,$	$+1\,$
1	$T_{1}(x)=\,$		$+x\,$
2	$T_{2}(x)=\,$	$-1\,$		$+2x^{2}\,$
3	$T_{3}(x)=\,$		$-3x\,$		$+4x^{3}\,$
4	$T_{4}(x)=\,$	$+1\,$		$-8x^{2}\,$		$+8x^{4}\,$
5	$T_{5}(x)=\,$		$+5x\,$		$-20x^{3}\,$		$+16x^{5}\,$
6	$T_{6}(x)=\,$	$-1\,$		$+18x^{2}\,$		$-48x^{4}\,$		$+32x^{6}\,$
7	$T_{7}(x)=\,$		$-7x\,$		$+56x^{3}\,$		$-112x^{5}\,$		$+64x^{7}\,$
8	$T_{8}(x)=\,$	$+1\,$		$-32x^{2}\,$		$+160x^{4}\,$		$-256x^{6}\,$		$+128x^{8}\,$
9	$T_{9}(x)=\,$		$+9x\,$		$-120x^{3}\,$		$+432x^{5}\,$		$-576x^{7}\,$		$+256x^{9}\,$
10	$T_{10}(x)=\,$	$-1\,$		$+50x^{2}\,$		$-400x^{4}\,$		$+1120x^{6}\,$		$-1280x^{8}\,$		$+512x^{10}\,$
11	$T_{11}(x)=\,$		$-11x\,$		$+220x^{3}\,$		$-1232x^{5}\,$		$+2816x^{7}\,$		$-2816x^{9}\,$		$+1024x^{11}\,$
12	$T_{12}(x)=\,$	$+1\,$		$-72x^{2}\,$		$+840x^{4}\,$		$-3584x^{6}\,$		$+6912x^{8}\,$		$-6144x^{10}\,$		$+2048x^{12}\,$

The triangle of coefficients of Chebyshev polynomials of the first kind $\scriptstyle T_{n}(x)\,$ gives the infinite sequence of finite sequences

{{1}, {1}, {-1, 2}, {-3, 4}, {1, -8, 8}, {5, -20, 16}, {-1, 18, -48, 32}, {-7, 56, -112, 64}, {1, -32, 160, -256, 128}, {9, -120, 432, -576, 256}, {-1, 50, -400, 1120, -1280, 512}, {-11, 220, -1232, 2816, -2816, 1024}, {1, -72, 840, -3584, 6912, -6144, 2048}, {13, -364, 2912, -9984, 16640, -13312, 4096}, ...}

Triangle of coefficients of Chebyshev polynomials of the first kind $\scriptstyle T_{n}(x)\,$. (Cf. A008310)

{1, 1, -1, 2, -3, 4, 1, -8, 8, 5, -20, 16, -1, 18, -48, 32, -7, 56, -112, 64, 1, -32, 160, -256, 128, 9, -120, 432, -576, 256, -1, 50, -400, 1120, -1280, 512, -11, 220, -1232, 2816, -2816, 1024, 1, -72, 840, -3584, 6912, -6144, 2048, 13, -364, 2912, -9984, 16640, -13312, 4096, ...}

Rows of coefficients of Chebyshev polynomials of the first kind

$T_{n,k}=\,?,\quad n\geq 0.\,$

Row sums of coefficients of Chebyshev polynomials of the first kind

$\sum _{k=0}^{n}T_{n,k}=1,\quad n\geq 0.\,$

Columns of coefficients of Chebyshev polynomials of the first kind

$T_{n,k}=\,?,\quad k\geq 0.\,$

Columns of absolute values of coefficients of Chebyshev polynomials of the first kind

Compare with the (2,1)-Pascal triangle columns.

Rising diagonals of coefficients of Chebyshev polynomials of the first kind

(...)

Rising diagonal sums of coefficients of Chebyshev polynomials of the first kind

Except for the 0^th rising diagonal, which sums to 1, the rising diagonal sums are all 0. Thus the j th rising diagonal sums to $\scriptstyle 0^{j},\,j\,\geq \,0\,$.

Falling diagonals of coefficients of Chebyshev polynomials of the first kind

(...)

Chebyshev polynomials of the second kind

The Chebyshev polynomials of the second kind are defined by the recurrence relation

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

The first few Chebyshev polynomials of the second kind are

$U_{0}(x)=1\,$

$U_{1}(x)=2x\,$

$U_{2}(x)=4x^{2}-1\,$

$U_{3}(x)=8x^{3}-4x\,$

$U_{4}(x)=16x^{4}-12x^{2}+1\,$

$U_{5}(x)=32x^{5}-32x^{3}+6x\,$

$U_{6}(x)=64x^{6}-80x^{4}+24x^{2}-1\,$

$U_{7}(x)=128x^{7}-192x^{5}+80x^{3}-8x\,$

$U_{8}(x)=256x^{8}-448x^{6}+240x^{4}-40x^{2}+1\,$

$U_{9}(x)=512x^{9}-1024x^{7}+672x^{5}-160x^{3}+10x.\,$

A closed-form formula (would be a Binet formula of the second type, except that the exponents are $\scriptstyle n+1\,$ instead of $\scriptstyle n\,$) (Cf. Fibonacci numbers#Binet's closed-form formula) giving the Chebyshev polynomials of the second kind is

$U_{n}(x)={\frac {(\rho _{+})^{n+1}-(\rho _{-})^{n+1}}{(\rho _{+})-(\rho _{-})}}={\frac {(x+{\sqrt {x^{2}-1}})^{n+1}-(x-{\sqrt {x^{2}-1}})^{n+1}}{(x+{\sqrt {x^{2}-1}})-(x-{\sqrt {x^{2}-1}})}}={\frac {(x+{\sqrt {x^{2}-1}})^{n+1}-(x-{\sqrt {x^{2}-1}})^{n+1}}{2{\sqrt {x^{2}-1}}}},\,$

where $\scriptstyle \rho _{+}\,$ and $\scriptstyle \rho _{-}\,$ are the roots of the quadratic polynomial in terms of $\scriptstyle t\,$

$1-2tx+t^{2}=0\,$

The ordinary generating function for $\scriptstyle U_{n}(x)\,$ is

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

The exponential generating function for $\scriptstyle U_{n}(x)\,$ is

$E_{\{U_{n}(x)\}}(t)=\sum _{n=0}^{\infty }U_{n}(x)~{\frac {t^{n}}{n!}}=\,?\,$

Triangle of coefficients of Chebyshev polynomials of the second kind

Chebyshev polynomials of the second kind

$n\,$	$U_{n}(x)=\,$	$a_{0}x^{0}\,$	$+a_{1}x^{1}\,$	$+a_{2}x^{2}\,$	$+a_{3}x^{3}\,$	$+a_{4}x^{4}\,$	$+a_{5}x^{5}\,$	$+a_{6}x^{6}\,$	$+a_{7}x^{7}\,$	$+a_{8}x^{8}\,$	$+a_{9}x^{9}\,$	$+a_{10}x^{10}\,$	$+a_{11}x^{11}\,$	$+a_{12}x^{12}\,$
0	$U_{0}(x)=\,$	$+1\,$
1	$U_{1}(x)=\,$		$+2x\,$
2	$U_{2}(x)=\,$	$-1\,$		$+4x^{2}\,$
3	$U_{3}(x)=\,$		$-4x\,$		$+8x^{3}\,$
4	$U_{4}(x)=\,$	$+1\,$		$-12x^{2}\,$		$+16x^{4}\,$
5	$U_{5}(x)=\,$		$+6x\,$		$-32x^{3}\,$		$+32x^{5}\,$
6	$U_{6}(x)=\,$	$-1\,$		$+24x^{2}\,$		$-80x^{4}\,$		$+64x^{6}\,$
7	$U_{7}(x)=\,$		$-8x\,$		$+80x^{3}\,$		$-192x^{5}\,$		$+128x^{7}\,$
8	$U_{8}(x)=\,$	$+1\,$		$-40x^{2}\,$		$+240x^{4}\,$		$-448x^{6}\,$		$+256x^{8}\,$
9	$U_{9}(x)=\,$		$+10x\,$		$-160x^{3}\,$		$+672x^{5}\,$		$-1024x^{7}\,$		$+512x^{9}\,$
10	$U_{10}(x)=\,$	$-1\,$		$+60x^{2}\,$		$-560x^{4}\,$		$+1792x^{6}\,$		$-2304x^{8}\,$		$+1024x^{10}\,$
11	$U_{11}(x)=\,$		$-12x\,$		$+280x^{3}\,$		$-1792x^{5}\,$		$+4608x^{7}\,$		$-5120x^{9}\,$		$+2048x^{11}\,$
12	$U_{12}(x)=\,$	$+1\,$		$-84x^{2}\,$		$+1120x^{4}\,$		$-5376x^{6}\,$		$+11520x^{8}\,$		$-11264x^{10}\,$		$+4096x^{12}\,$

The triangle of coefficients of Chebyshev polynomials $\scriptstyle U_{n}(x)\,$ gives the infinite sequence of finite sequences

{{1}, {2}, {-1, 4}, {-4, 8}, {1, -12, 16}, {6, -32, 32}, {-1, 24, -80, 64}, {-8, 80, -192, 128}, {1, -40, 240, -448, 256}, {10, -160, 672, -1024, 512}, {-1, 60, -560, 1792, -2304, 1024}, {-12, 280, -1792, 4608, -5120, 2048}, {1, -84, 1120, -5376, 11520, -11264, 4096}, ...}

Triangle of coefficients of Chebyshev polynomials $\scriptstyle U_{n}(x)\,$. (Cf. A008312)

{1, 2, -1, 4, -4, 8, 1, -12, 16, 6, -32, 32, -1, 24, -80, 64, -8, 80, -192, 128, 1, -40, 240, -448, 256, 10, -160, 672, -1024, 512, -1, 60, -560, 1792, -2304, 1024, -12, 280, -1792, 4608, -5120, 2048, 1, -84, 1120, -5376, 11520, -11264, 4096, ...}

Rows of coefficients of Chebyshev polynomials of the second kind

$U_{n,k}=\,?,\quad n\geq 0.\,$

Row sums of coefficients of Chebyshev polynomials of the second kind

$\sum _{k=0}^{n}U_{n,k}=n+1,\quad n\geq 0.\,$

Columns of coefficients of Chebyshev polynomials of the second kind

$U_{n,k}=\,?,\quad k\geq 0.\,$

Columns of absolute values of coefficients of Chebyshev polynomials of the second kind

Compare with the (1,1)-Pascal triangle columns.

Rising diagonals of coefficients of Chebyshev polynomials of the second kind

(...)

Rising diagonal sums of coefficients of Chebyshev polynomials of the second kind

The rising diagonal sums are alternatively 1 and 0.

Falling diagonals of coefficients of Chebyshev polynomials of the second kind

(...)

See also

• Special cases of the Chebyshev polynomials