Fungrim entry: 85e42e

Table of

T_{n}\!\left(x\right)

for

0 \le n \le 15

$n$	$T_{n}\!\left(x\right)$
0	1
1	$x$
2	$2 {x}^{2} - 1$
3	$4 {x}^{3} - 3 x$
4	$8 {x}^{4} - 8 {x}^{2} + 1$
5	$16 {x}^{5} - 20 {x}^{3} + 5 x$
6	$32 {x}^{6} - 48 {x}^{4} + 18 {x}^{2} - 1$
7	$64 {x}^{7} - 112 {x}^{5} + 56 {x}^{3} - 7 x$
8	$128 {x}^{8} - 256 {x}^{6} + 160 {x}^{4} - 32 {x}^{2} + 1$
9	$256 {x}^{9} - 576 {x}^{7} + 432 {x}^{5} - 120 {x}^{3} + 9 x$
10	$512 {x}^{10} - 1280 {x}^{8} + 1120 {x}^{6} - 400 {x}^{4} + 50 {x}^{2} - 1$
11	$1024 {x}^{11} - 2816 {x}^{9} + 2816 {x}^{7} - 1232 {x}^{5} + 220 {x}^{3} - 11 x$
12	$2048 {x}^{12} - 6144 {x}^{10} + 6912 {x}^{8} - 3584 {x}^{6} + 840 {x}^{4} - 72 {x}^{2} + 1$
13	$4096 {x}^{13} - 13312 {x}^{11} + 16640 {x}^{9} - 9984 {x}^{7} + 2912 {x}^{5} - 364 {x}^{3} + 13 x$
14	$8192 {x}^{14} - 28672 {x}^{12} + 39424 {x}^{10} - 26880 {x}^{8} + 9408 {x}^{6} - 1568 {x}^{4} + 98 {x}^{2} - 1$
15	$16384 {x}^{15} - 61440 {x}^{13} + 92160 {x}^{11} - 70400 {x}^{9} + 28800 {x}^{7} - 6048 {x}^{5} + 560 {x}^{3} - 15 x$

Table data:

\left(n, p\right)

such that

T_{n}\!\left(x\right) = p

Assumptions:

x \in \mathbb{C}

TeX:

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("85e42e"),
    Description("Table of", ChebyshevT(n, x), "for", LessEqual(0, n, 15)),
    Table(TableRelation(Tuple(n, p), Equal(ChebyshevT(n, x), p)), TableHeadings(n, ChebyshevT(n, x)), TableSplit(1), List(Tuple(0, 1), Tuple(1, x), Tuple(2, Sub(Mul(2, Pow(x, 2)), 1)), Tuple(3, Sub(Mul(4, Pow(x, 3)), Mul(3, x))), Tuple(4, Add(Sub(Mul(8, Pow(x, 4)), Mul(8, Pow(x, 2))), 1)), Tuple(5, Add(Sub(Mul(16, Pow(x, 5)), Mul(20, Pow(x, 3))), Mul(5, x))), Tuple(6, Sub(Add(Sub(Mul(32, Pow(x, 6)), Mul(48, Pow(x, 4))), Mul(18, Pow(x, 2))), 1)), Tuple(7, Sub(Add(Sub(Mul(64, Pow(x, 7)), Mul(112, Pow(x, 5))), Mul(56, Pow(x, 3))), Mul(7, x))), Tuple(8, Add(Sub(Add(Sub(Mul(128, Pow(x, 8)), Mul(256, Pow(x, 6))), Mul(160, Pow(x, 4))), Mul(32, Pow(x, 2))), 1)), Tuple(9, Add(Sub(Add(Sub(Mul(256, Pow(x, 9)), Mul(576, Pow(x, 7))), Mul(432, Pow(x, 5))), Mul(120, Pow(x, 3))), Mul(9, x))), Tuple(10, Sub(Add(Sub(Add(Sub(Mul(512, Pow(x, 10)), Mul(1280, Pow(x, 8))), Mul(1120, Pow(x, 6))), Mul(400, Pow(x, 4))), Mul(50, Pow(x, 2))), 1)), Tuple(11, Sub(Add(Sub(Add(Sub(Mul(1024, Pow(x, 11)), Mul(2816, Pow(x, 9))), Mul(2816, Pow(x, 7))), Mul(1232, Pow(x, 5))), Mul(220, Pow(x, 3))), Mul(11, x))), Tuple(12, Add(Sub(Add(Sub(Add(Sub(Mul(2048, Pow(x, 12)), Mul(6144, Pow(x, 10))), Mul(6912, Pow(x, 8))), Mul(3584, Pow(x, 6))), Mul(840, Pow(x, 4))), Mul(72, Pow(x, 2))), 1)), Tuple(13, Add(Sub(Add(Sub(Add(Sub(Mul(4096, Pow(x, 13)), Mul(13312, Pow(x, 11))), Mul(16640, Pow(x, 9))), Mul(9984, Pow(x, 7))), Mul(2912, Pow(x, 5))), Mul(364, Pow(x, 3))), Mul(13, x))), Tuple(14, Sub(Add(Sub(Add(Sub(Add(Sub(Mul(8192, Pow(x, 14)), Mul(28672, Pow(x, 12))), Mul(39424, Pow(x, 10))), Mul(26880, Pow(x, 8))), Mul(9408, Pow(x, 6))), Mul(1568, Pow(x, 4))), Mul(98, Pow(x, 2))), 1)), Tuple(15, Sub(Add(Sub(Add(Sub(Add(Sub(Mul(16384, Pow(x, 15)), Mul(61440, Pow(x, 13))), Mul(92160, Pow(x, 11))), Mul(70400, Pow(x, 9))), Mul(28800, Pow(x, 7))), Mul(6048, Pow(x, 5))), Mul(560, Pow(x, 3))), Mul(15, x))))),
    Variables(x),
    Assumptions(Element(x, CC)))

Topics using this entry

Chebyshev polynomials