Fungrim entry: fd8310

Table of

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

for

0 \le n \le 15

$n$	$U_{n}\!\left(x\right)$
0	1
1	$2 x$
2	$4 {x}^{2} - 1$
3	$8 {x}^{3} - 4 x$
4	$16 {x}^{4} - 12 {x}^{2} + 1$
5	$32 {x}^{5} - 32 {x}^{3} + 6 x$
6	$64 {x}^{6} - 80 {x}^{4} + 24 {x}^{2} - 1$
7	$128 {x}^{7} - 192 {x}^{5} + 80 {x}^{3} - 8 x$
8	$256 {x}^{8} - 448 {x}^{6} + 240 {x}^{4} - 40 {x}^{2} + 1$
9	$512 {x}^{9} - 1024 {x}^{7} + 672 {x}^{5} - 160 {x}^{3} + 10 x$
10	$1024 {x}^{10} - 2304 {x}^{8} + 1792 {x}^{6} - 560 {x}^{4} + 60 {x}^{2} - 1$
11	$2048 {x}^{11} - 5120 {x}^{9} + 4608 {x}^{7} - 1792 {x}^{5} + 280 {x}^{3} - 12 x$
12	$4096 {x}^{12} - 11264 {x}^{10} + 11520 {x}^{8} - 5376 {x}^{6} + 1120 {x}^{4} - 84 {x}^{2} + 1$
13	$8192 {x}^{13} - 24576 {x}^{11} + 28160 {x}^{9} - 15360 {x}^{7} + 4032 {x}^{5} - 448 {x}^{3} + 14 x$
14	$16384 {x}^{14} - 53248 {x}^{12} + 67584 {x}^{10} - 42240 {x}^{8} + 13440 {x}^{6} - 2016 {x}^{4} + 112 {x}^{2} - 1$
15	$32768 {x}^{15} - 114688 {x}^{13} + 159744 {x}^{11} - 112640 {x}^{9} + 42240 {x}^{7} - 8064 {x}^{5} + 672 {x}^{3} - 16 x$

Table data:

\left(n, p\right)

such that

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

Assumptions:

x \in \mathbb{C}

TeX:

x \in \mathbb{C}

Definitions:

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

Source code for this entry:

Entry(ID("fd8310"),
    Description("Table of", ChebyshevU(n, x), "for", LessEqual(0, n, 15)),
    Table(TableRelation(Tuple(n, p), Equal(ChebyshevU(n, x), p)), TableHeadings(n, ChebyshevU(n, x)), TableSplit(1), List(Tuple(0, 1), Tuple(1, Mul(2, x)), Tuple(2, Sub(Mul(4, Pow(x, 2)), 1)), Tuple(3, Sub(Mul(8, Pow(x, 3)), Mul(4, x))), Tuple(4, Add(Sub(Mul(16, Pow(x, 4)), Mul(12, Pow(x, 2))), 1)), Tuple(5, Add(Sub(Mul(32, Pow(x, 5)), Mul(32, Pow(x, 3))), Mul(6, x))), Tuple(6, Sub(Add(Sub(Mul(64, Pow(x, 6)), Mul(80, Pow(x, 4))), Mul(24, Pow(x, 2))), 1)), Tuple(7, Sub(Add(Sub(Mul(128, Pow(x, 7)), Mul(192, Pow(x, 5))), Mul(80, Pow(x, 3))), Mul(8, x))), Tuple(8, Add(Sub(Add(Sub(Mul(256, Pow(x, 8)), Mul(448, Pow(x, 6))), Mul(240, Pow(x, 4))), Mul(40, Pow(x, 2))), 1)), Tuple(9, Add(Sub(Add(Sub(Mul(512, Pow(x, 9)), Mul(1024, Pow(x, 7))), Mul(672, Pow(x, 5))), Mul(160, Pow(x, 3))), Mul(10, x))), Tuple(10, Sub(Add(Sub(Add(Sub(Mul(1024, Pow(x, 10)), Mul(2304, Pow(x, 8))), Mul(1792, Pow(x, 6))), Mul(560, Pow(x, 4))), Mul(60, Pow(x, 2))), 1)), Tuple(11, Sub(Add(Sub(Add(Sub(Mul(2048, Pow(x, 11)), Mul(5120, Pow(x, 9))), Mul(4608, Pow(x, 7))), Mul(1792, Pow(x, 5))), Mul(280, Pow(x, 3))), Mul(12, x))), Tuple(12, Add(Sub(Add(Sub(Add(Sub(Mul(4096, Pow(x, 12)), Mul(11264, Pow(x, 10))), Mul(11520, Pow(x, 8))), Mul(5376, Pow(x, 6))), Mul(1120, Pow(x, 4))), Mul(84, Pow(x, 2))), 1)), Tuple(13, Add(Sub(Add(Sub(Add(Sub(Mul(8192, Pow(x, 13)), Mul(24576, Pow(x, 11))), Mul(28160, Pow(x, 9))), Mul(15360, Pow(x, 7))), Mul(4032, Pow(x, 5))), Mul(448, Pow(x, 3))), Mul(14, x))), Tuple(14, Sub(Add(Sub(Add(Sub(Add(Sub(Mul(16384, Pow(x, 14)), Mul(53248, Pow(x, 12))), Mul(67584, Pow(x, 10))), Mul(42240, Pow(x, 8))), Mul(13440, Pow(x, 6))), Mul(2016, Pow(x, 4))), Mul(112, Pow(x, 2))), 1)), Tuple(15, Sub(Add(Sub(Add(Sub(Add(Sub(Mul(32768, Pow(x, 15)), Mul(114688, Pow(x, 13))), Mul(159744, Pow(x, 11))), Mul(112640, Pow(x, 9))), Mul(42240, Pow(x, 7))), Mul(8064, Pow(x, 5))), Mul(672, Pow(x, 3))), Mul(16, x))))),
    Variables(x),
    Assumptions(Element(x, CC)))

Topics using this entry

Chebyshev polynomials