Chebyshev polynomials

Symbol: ChebyshevT —

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

— Chebyshev polynomial of the first kind

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind

Source code for this entry:

Entry(ID("1a0c43"),
    SymbolDefinition(ChebyshevT, ChebyshevT(n, x), "Chebyshev polynomial of the first kind"))

d4e9aa

Symbol: ChebyshevU —

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

— Chebyshev polynomial of the second kind

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind

Source code for this entry:

Entry(ID("d4e9aa"),
    SymbolDefinition(ChebyshevU, ChebyshevU(n, x), "Chebyshev polynomial of the second kind"))

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)))

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)))

c76e72

T_{0}\!\left(x\right) = 1

Assumptions:

x \in \mathbb{C}

TeX:

T_{0}\!\left(x\right) = 1

x \in \mathbb{C}

Definitions:

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

Source code for this entry:

Entry(ID("c76e72"),
    Formula(Equal(ChebyshevT(0, x), 1)),
    Variables(x),
    Assumptions(Element(x, CC)))

be5652

T_{1}\!\left(x\right) = x

Assumptions:

x \in \mathbb{C}

TeX:

T_{1}\!\left(x\right) = x

x \in \mathbb{C}

Definitions:

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

Source code for this entry:

Entry(ID("be5652"),
    Formula(Equal(ChebyshevT(1, x), x)),
    Variables(x),
    Assumptions(Element(x, CC)))

48765b

U_{0}\!\left(x\right) = 1

Assumptions:

x \in \mathbb{C}

TeX:

U_{0}\!\left(x\right) = 1

x \in \mathbb{C}

Definitions:

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

Source code for this entry:

Entry(ID("48765b"),
    Formula(Equal(ChebyshevU(0, x), 1)),
    Variables(x),
    Assumptions(Element(x, CC)))

75eacb

U_{1}\!\left(x\right) = 2 x

Assumptions:

x \in \mathbb{C}

TeX:

U_{1}\!\left(x\right) = 2 x

x \in \mathbb{C}

Definitions:

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

Source code for this entry:

Entry(ID("75eacb"),
    Formula(Equal(ChebyshevU(1, x), Mul(2, x))),
    Variables(x),
    Assumptions(Element(x, CC)))

9001e6

U_{-1}\!\left(x\right) = 0

Assumptions:

x \in \mathbb{C}

TeX:

U_{-1}\!\left(x\right) = 0

x \in \mathbb{C}

Definitions:

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

Source code for this entry:

Entry(ID("9001e6"),
    Formula(Equal(ChebyshevU(-1, x), 0)),
    Variables(x),
    Assumptions(Element(x, CC)))

fc5d42

T_{n}\!\left(1\right) = 1

Assumptions:

n \in \mathbb{Z}

TeX:

T_{n}\!\left(1\right) = 1

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("fc5d42"),
    Formula(Equal(ChebyshevT(n, 1), 1)),
    Variables(n),
    Assumptions(Element(n, ZZ)))

2760e7

T_{n}\!\left(-1\right) = {\left(-1\right)}^{n}

Assumptions:

n \in \mathbb{Z}

TeX:

T_{n}\!\left(-1\right) = {\left(-1\right)}^{n}

n \in \mathbb{Z}

Definitions:

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

Source code for this entry:

Entry(ID("2760e7"),
    Formula(Equal(ChebyshevT(n, -1), Pow(-1, n))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

a46d91

T_{2 n}\!\left(0\right) = {\left(-1\right)}^{n}

Assumptions:

n \in \mathbb{Z}

TeX:

T_{2 n}\!\left(0\right) = {\left(-1\right)}^{n}

n \in \mathbb{Z}

Definitions:

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

Source code for this entry:

Entry(ID("a46d91"),
    Formula(Equal(ChebyshevT(Mul(2, n), 0), Pow(-1, n))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

42102c

T_{2 n + 1}\!\left(0\right) = 0

Assumptions:

n \in \mathbb{Z}

TeX:

T_{2 n + 1}\!\left(0\right) = 0

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("42102c"),
    Formula(Equal(ChebyshevT(Add(Mul(2, n), 1), 0), 0)),
    Variables(n),
    Assumptions(Element(n, ZZ)))

e03fa4

U_{n}\!\left(1\right) = n + 1

Assumptions:

n \in \mathbb{Z}

TeX:

U_{n}\!\left(1\right) = n + 1

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("e03fa4"),
    Formula(Equal(ChebyshevU(n, 1), Add(n, 1))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

be9a45

U_{n}\!\left(-1\right) = {\left(-1\right)}^{n} \left(n + 1\right)

Assumptions:

n \in \mathbb{Z}

TeX:

U_{n}\!\left(-1\right) = {\left(-1\right)}^{n} \left(n + 1\right)

n \in \mathbb{Z}

Definitions:

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

Source code for this entry:

Entry(ID("be9a45"),
    Formula(Equal(ChebyshevU(n, -1), Mul(Pow(-1, n), Add(n, 1)))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

2a5337

U_{2 n}\!\left(0\right) = {\left(-1\right)}^{n}

Assumptions:

n \in \mathbb{Z}

TeX:

U_{2 n}\!\left(0\right) = {\left(-1\right)}^{n}

n \in \mathbb{Z}

Definitions:

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

Source code for this entry:

Entry(ID("2a5337"),
    Formula(Equal(ChebyshevU(Mul(2, n), 0), Pow(-1, n))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

7d111e

U_{2 n + 1}\!\left(0\right) = 0

Assumptions:

n \in \mathbb{Z}

TeX:

U_{2 n + 1}\!\left(0\right) = 0

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("7d111e"),
    Formula(Equal(ChebyshevU(Add(Mul(2, n), 1), 0), 0)),
    Variables(n),
    Assumptions(Element(n, ZZ)))

7a7d1d

\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} T_{n}\!\left(x\right) = \left\{ \cos\!\left(\frac{2 k - 1}{2 n} \pi\right) : k \in \{1, 2, \ldots, n\} \right\}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} T_{n}\!\left(x\right) = \left\{ \cos\!\left(\frac{2 k - 1}{2 n} \pi\right) : k \in \{1, 2, \ldots, n\} \right\}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`CC`	$\mathbb{C}$	Complex numbers
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("7a7d1d"),
    Formula(Equal(Zeros(ChebyshevT(n, x), ForElement(x, CC)), Set(Cos(Mul(Div(Sub(Mul(2, k), 1), Mul(2, n)), Pi)), ForElement(k, Range(1, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

ce39ac

\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} U_{n}\!\left(x\right) = \left\{ \cos\!\left(\frac{k}{n + 1} \pi\right) : k \in \{1, 2, \ldots, n\} \right\}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} U_{n}\!\left(x\right) = \left\{ \cos\!\left(\frac{k}{n + 1} \pi\right) : k \in \{1, 2, \ldots, n\} \right\}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`CC`	$\mathbb{C}$	Complex numbers
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("ce39ac"),
    Formula(Equal(Zeros(ChebyshevU(n, x), ForElement(x, CC)), Set(Cos(Mul(Div(k, Add(n, 1)), Pi)), ForElement(k, Range(1, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

3d25dd

\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) \in \left\{-1, 1\right\}\right] = \left\{ \cos\!\left(\frac{k}{n} \pi\right) : k \in \{0, 1, \ldots, n\} \right\}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) \in \left\{-1, 1\right\}\right] = \left\{ \cos\!\left(\frac{k}{n} \pi\right) : k \in \{0, 1, \ldots, n\} \right\}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Solutions`	$\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x)$	Solution set
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`CC`	$\mathbb{C}$	Complex numbers
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("3d25dd"),
    Formula(Equal(Solutions(Brackets(Element(ChebyshevT(n, x), Set(-1, 1))), ForElement(x, CC)), Set(Cos(Mul(Div(k, n), Pi)), ForElement(k, Range(0, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

b5a25e

\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) = 1\right] = \left\{ \cos\!\left(\frac{2 k}{n} \pi\right) : k \in \{0, 1, \ldots, \left\lfloor \frac{n}{2} \right\rfloor\} \right\}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) = 1\right] = \left\{ \cos\!\left(\frac{2 k}{n} \pi\right) : k \in \{0, 1, \ldots, \left\lfloor \frac{n}{2} \right\rfloor\} \right\}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Solutions`	$\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x)$	Solution set
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`CC`	$\mathbb{C}$	Complex numbers
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("b5a25e"),
    Formula(Equal(Solutions(Brackets(Equal(ChebyshevT(n, x), 1)), ForElement(x, CC)), Set(Cos(Mul(Div(Mul(2, k), n), Pi)), ForElement(k, Range(0, Floor(Div(n, 2))))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

db2b0a

\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) = -1\right] = \left\{ \cos\!\left(\frac{2 k - 1}{n} \pi\right) : k \in \{1, 2, \ldots, \left\lfloor \frac{n + 1}{2} \right\rfloor\} \right\}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) = -1\right] = \left\{ \cos\!\left(\frac{2 k - 1}{n} \pi\right) : k \in \{1, 2, \ldots, \left\lfloor \frac{n + 1}{2} \right\rfloor\} \right\}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Solutions`	$\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x)$	Solution set
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`CC`	$\mathbb{C}$	Complex numbers
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("db2b0a"),
    Formula(Equal(Solutions(Brackets(Equal(ChebyshevT(n, x), -1)), ForElement(x, CC)), Set(Cos(Mul(Div(Sub(Mul(2, k), 1), n), Pi)), ForElement(k, Range(1, Floor(Div(Add(n, 1), 2))))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

6a24ab

T_{n}\!\left(-x\right) = {\left(-1\right)}^{n} T_{n}\!\left(x\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(-x\right) = {\left(-1\right)}^{n} T_{n}\!\left(x\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; 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
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("6a24ab"),
    Formula(Equal(ChebyshevT(n, Neg(x)), Mul(Pow(-1, n), ChebyshevT(n, x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

88aeb6

U_{n}\!\left(-x\right) = {\left(-1\right)}^{n} U_{n}\!\left(x\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{n}\!\left(-x\right) = {\left(-1\right)}^{n} U_{n}\!\left(x\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; 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
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("88aeb6"),
    Formula(Equal(ChebyshevU(n, Neg(x)), Mul(Pow(-1, n), ChebyshevU(n, x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

9093a3

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

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

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

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9093a3"),
    Formula(Equal(ChebyshevT(Neg(n), x), ChebyshevT(n, x))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

78f5bb

U_{-n}\!\left(x\right) = -U_{n - 2}\!\left(x\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{-n}\!\left(x\right) = -U_{n - 2}\!\left(x\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("78f5bb"),
    Formula(Equal(ChebyshevU(Neg(n), x), Neg(ChebyshevU(Sub(n, 2), x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

2c26a1

\int_{-1}^{1} T_{n}\!\left(x\right) T_{m}\!\left(x\right) \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = \begin{cases} 0, & n \ne m\\\pi, & n = m = 0\\\frac{\pi}{2}, & n = m \;\mathbin{\operatorname{and}}\; n \ne 0\\ \end{cases}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}

TeX:

\int_{-1}^{1} T_{n}\!\left(x\right) T_{m}\!\left(x\right) \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = \begin{cases} 0, & n \ne m\\\pi, & n = m = 0\\\frac{\pi}{2}, & n = m \;\mathbin{\operatorname{and}}\; n \ne 0\\ \end{cases}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("2c26a1"),
    Formula(Equal(Integral(Mul(Mul(ChebyshevT(n, x), ChebyshevT(m, x)), Div(1, Sqrt(Sub(1, Pow(x, 2))))), For(x, -1, 1)), Cases(Tuple(0, NotEqual(n, m)), Tuple(Pi, Equal(n, m, 0)), Tuple(Div(Pi, 2), And(Equal(n, m), NotEqual(n, 0)))))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)))))

473c36

\int_{-1}^{1} U_{n}\!\left(x\right) U_{m}\!\left(x\right) \sqrt{1 - {x}^{2}} \, dx = \begin{cases} 0, & n \ne m\\\frac{\pi}{2}, & n = m\\ \end{cases}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}

TeX:

\int_{-1}^{1} U_{n}\!\left(x\right) U_{m}\!\left(x\right) \sqrt{1 - {x}^{2}} \, dx = \begin{cases} 0, & n \ne m\\\frac{\pi}{2}, & n = m\\ \end{cases}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("473c36"),
    Formula(Equal(Integral(Mul(Mul(ChebyshevU(n, x), ChebyshevU(m, x)), Sqrt(Sub(1, Pow(x, 2)))), For(x, -1, 1)), Cases(Tuple(0, NotEqual(n, m)), Tuple(Div(Pi, 2), Equal(n, m))))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)))))

3d77ab

\int_{-1}^{1} T_{n}\!\left(x\right) {x}^{m} \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = 0

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \{0, 1, \ldots, n - 1\}

TeX:

\int_{-1}^{1} T_{n}\!\left(x\right) {x}^{m} \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = 0

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \{0, 1, \ldots, n - 1\}

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints

Source code for this entry:

Entry(ID("3d77ab"),
    Formula(Equal(Integral(Mul(Mul(ChebyshevT(n, x), Pow(x, m)), Div(1, Sqrt(Sub(1, Pow(x, 2))))), For(x, -1, 1)), 0)),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, Range(0, Sub(n, 1))))))

0ed026

\left(1 - {x}^{2}\right) y''(x) - x y'(x) + {n}^{2} y(x) = 0\; \text{ where } y(x) = {c}_{1} T_{n}\!\left(x\right) + {c}_{2} U_{n - 1}\!\left(x\right) \sqrt{1 - {x}^{2}}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left({c}_{2} = 0 \;\mathbin{\operatorname{or}}\; x \notin \left(-\infty, 1\right] \cup \left[1, \infty\right)\right)

TeX:

\left(1 - {x}^{2}\right) y''(x) - x y'(x) + {n}^{2} y(x) = 0\; \text{ where } y(x) = {c}_{1} T_{n}\!\left(x\right) + {c}_{2} U_{n - 1}\!\left(x\right) \sqrt{1 - {x}^{2}}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left({c}_{2} = 0 \;\mathbin{\operatorname{or}}\; x \notin \left(-\infty, 1\right] \cup \left[1, \infty\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval

Source code for this entry:

Entry(ID("0ed026"),
    Formula(Where(Equal(Add(Sub(Mul(Sub(1, Pow(x, 2)), ComplexDerivative(y(x), For(x, x, 2))), Mul(x, ComplexDerivative(y(x), For(x, x, 1)))), Mul(Pow(n, 2), y(x))), 0), Equal(y(x), Add(Mul(Subscript(c, 1), ChebyshevT(n, x)), Mul(Mul(Subscript(c, 2), ChebyshevU(Sub(n, 1), x)), Sqrt(Sub(1, Pow(x, 2)))))))),
    Variables(n, x, Subscript(c, 1), Subscript(c, 2)),
    Assumptions(And(Element(n, ZZ), Element(x, CC), Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC), Or(Equal(Subscript(c, 2), 0), NotElement(x, Union(OpenClosedInterval(Neg(Infinity), 1), ClosedOpenInterval(1, Infinity)))))))

30b67b

\left(1 - {x}^{2}\right) y''(x) - 3 x y'(x) + n \left(n + 2\right) y(x) = 0\; \text{ where } y(x) = {c}_{1} U_{n}\!\left(x\right) + {c}_{2} \frac{T_{n + 1}\!\left(x\right)}{\sqrt{1 - {x}^{2}}}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left({c}_{2} = 0 \;\mathbin{\operatorname{or}}\; x \notin \left(-\infty, 1\right] \cup \left[1, \infty\right)\right) \;\mathbin{\operatorname{and}}\; x \notin \left\{-1, 1\right\}

TeX:

\left(1 - {x}^{2}\right) y''(x) - 3 x y'(x) + n \left(n + 2\right) y(x) = 0\; \text{ where } y(x) = {c}_{1} U_{n}\!\left(x\right) + {c}_{2} \frac{T_{n + 1}\!\left(x\right)}{\sqrt{1 - {x}^{2}}}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{1} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; {c}_{2} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left({c}_{2} = 0 \;\mathbin{\operatorname{or}}\; x \notin \left(-\infty, 1\right] \cup \left[1, \infty\right)\right) \;\mathbin{\operatorname{and}}\; x \notin \left\{-1, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval

Source code for this entry:

Entry(ID("30b67b"),
    Formula(Where(Equal(Add(Sub(Mul(Sub(1, Pow(x, 2)), ComplexDerivative(y(x), For(x, x, 2))), Mul(Mul(3, x), ComplexDerivative(y(x), For(x, x, 1)))), Mul(Mul(n, Add(n, 2)), y(x))), 0), Equal(y(x), Add(Mul(Subscript(c, 1), ChebyshevU(n, x)), Mul(Subscript(c, 2), Div(ChebyshevT(Add(n, 1), x), Sqrt(Sub(1, Pow(x, 2))))))))),
    Variables(n, x, Subscript(c, 1), Subscript(c, 2)),
    Assumptions(And(Element(n, ZZ), Element(x, CC), Element(Subscript(c, 1), CC), Element(Subscript(c, 2), CC), Or(Equal(Subscript(c, 2), 0), NotElement(x, Union(OpenClosedInterval(Neg(Infinity), 1), ClosedOpenInterval(1, Infinity)))), NotElement(x, Set(-1, 1)))))

faeed9

T_{n}\!\left(x\right) = 2 x T_{n - 1}\!\left(x\right) - T_{n - 2}\!\left(x\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = 2 x T_{n - 1}\!\left(x\right) - T_{n - 2}\!\left(x\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("faeed9"),
    Formula(Equal(ChebyshevT(n, x), Sub(Mul(Mul(2, x), ChebyshevT(Sub(n, 1), x)), ChebyshevT(Sub(n, 2), x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

d1ef91

U_{n}\!\left(x\right) = 2 x U_{n - 1}\!\left(x\right) - U_{n - 2}\!\left(x\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{n}\!\left(x\right) = 2 x U_{n - 1}\!\left(x\right) - U_{n - 2}\!\left(x\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d1ef91"),
    Formula(Equal(ChebyshevU(n, x), Sub(Mul(Mul(2, x), ChebyshevU(Sub(n, 1), x)), ChebyshevU(Sub(n, 2), x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

8a785a

T_{n}\!\left(x\right) = 2 x T_{n + 1}\!\left(x\right) - T_{n + 2}\!\left(x\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = 2 x T_{n + 1}\!\left(x\right) - T_{n + 2}\!\left(x\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("8a785a"),
    Formula(Equal(ChebyshevT(n, x), Sub(Mul(Mul(2, x), ChebyshevT(Add(n, 1), x)), ChebyshevT(Add(n, 2), x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

303204

U_{n}\!\left(x\right) = 2 x U_{n + 1}\!\left(x\right) - U_{n + 2}\!\left(x\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{n}\!\left(x\right) = 2 x U_{n + 1}\!\left(x\right) - U_{n + 2}\!\left(x\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("303204"),
    Formula(Equal(ChebyshevU(n, x), Sub(Mul(Mul(2, x), ChebyshevU(Add(n, 1), x)), ChebyshevU(Add(n, 2), x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

7b2c26

T_{n}\!\left(x\right) = x T_{n - 1}\!\left(x\right) - \left(1 - {x}^{2}\right) U_{n - 2}\!\left(x\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = x T_{n - 1}\!\left(x\right) - \left(1 - {x}^{2}\right) U_{n - 2}\!\left(x\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; 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
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7b2c26"),
    Formula(Equal(ChebyshevT(n, x), Sub(Mul(x, ChebyshevT(Sub(n, 1), x)), Mul(Sub(1, Pow(x, 2)), ChebyshevU(Sub(n, 2), x))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

ce5e03

U_{n}\!\left(x\right) = x U_{n - 1}\!\left(x\right) + T_{n}\!\left(x\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{n}\!\left(x\right) = x U_{n - 1}\!\left(x\right) + T_{n}\!\left(x\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("ce5e03"),
    Formula(Equal(ChebyshevU(n, x), Add(Mul(x, ChebyshevU(Sub(n, 1), x)), ChebyshevT(n, x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

0649c9

T_{n}\!\left(x\right) = \frac{U_{n}\!\left(x\right) - U_{n - 2}\!\left(x\right)}{2}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = \frac{U_{n}\!\left(x\right) - U_{n - 2}\!\left(x\right)}{2}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0649c9"),
    Formula(Equal(ChebyshevT(n, x), Div(Sub(ChebyshevU(n, x), ChebyshevU(Sub(n, 2), x)), 2))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

844561

T_{n}\!\left(x\right) = U_{n}\!\left(x\right) - x U_{n - 1}\!\left(x\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = U_{n}\!\left(x\right) - x U_{n - 1}\!\left(x\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("844561"),
    Formula(Equal(ChebyshevT(n, x), Sub(ChebyshevU(n, x), Mul(x, ChebyshevU(Sub(n, 1), x))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

7e882c

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

Assumptions:

m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

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

m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7e882c"),
    Formula(Equal(ChebyshevT(m, ChebyshevT(n, x)), ChebyshevT(Mul(m, n), x))),
    Variables(m, n, x),
    Assumptions(And(Element(m, ZZ), Element(n, ZZ), Element(x, CC))))

ed5222

T_{m}\!\left(x\right) T_{n}\!\left(x\right) = \frac{T_{m + n}\!\left(x\right) + T_{\left|m - n\right|}\!\left(x\right)}{2}

Assumptions:

m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{m}\!\left(x\right) T_{n}\!\left(x\right) = \frac{T_{m + n}\!\left(x\right) + T_{\left|m - n\right|}\!\left(x\right)}{2}

m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Abs`	$\left\|z\right\|$	Absolute value
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("ed5222"),
    Formula(Equal(Mul(ChebyshevT(m, x), ChebyshevT(n, x)), Div(Add(ChebyshevT(Add(m, n), x), ChebyshevT(Abs(Sub(m, n)), x)), 2))),
    Variables(m, n, x),
    Assumptions(And(Element(m, ZZ), Element(n, ZZ), Element(x, CC))))

4b83c6

T_{2 n}\!\left(x\right) = 2 T_{n}^{2}\!\left(x\right) - 1

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{2 n}\!\left(x\right) = 2 T_{n}^{2}\!\left(x\right) - 1

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; 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
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4b83c6"),
    Formula(Equal(ChebyshevT(Mul(2, n), x), Sub(Mul(2, Pow(ChebyshevT(n, x), 2)), 1))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

de0968

T_{2 n + 1}\!\left(x\right) = 2 T_{n + 1}\!\left(x\right) T_{n}\!\left(x\right) - x

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{2 n + 1}\!\left(x\right) = 2 T_{n + 1}\!\left(x\right) T_{n}\!\left(x\right) - x

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("de0968"),
    Formula(Equal(ChebyshevT(Add(Mul(2, n), 1), x), Sub(Mul(Mul(2, ChebyshevT(Add(n, 1), x)), ChebyshevT(n, x)), x))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

82288c

T_{2 n}\!\left(x\right) = T_{n}\!\left(2 {x}^{2} - 1\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{2 n}\!\left(x\right) = T_{n}\!\left(2 {x}^{2} - 1\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; 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
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("82288c"),
    Formula(Equal(ChebyshevT(Mul(2, n), x), ChebyshevT(n, Sub(Mul(2, Pow(x, 2)), 1)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

5f09f4

U_{2 n}\!\left(x\right) = T_{n}\!\left(2 {x}^{2} - 1\right) + U_{n - 1}\!\left(2 {x}^{2} - 1\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{2 n}\!\left(x\right) = T_{n}\!\left(2 {x}^{2} - 1\right) + U_{n - 1}\!\left(2 {x}^{2} - 1\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

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

Source code for this entry:

Entry(ID("5f09f4"),
    Formula(Equal(ChebyshevU(Mul(2, n), x), Add(ChebyshevT(n, Sub(Mul(2, Pow(x, 2)), 1)), ChebyshevU(Sub(n, 1), Sub(Mul(2, Pow(x, 2)), 1))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

fda800

T_{n}\!\left(x\right) = \cos\!\left(n \operatorname{acos}(x)\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = \cos\!\left(n \operatorname{acos}(x)\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Cos`	$\cos(z)$	Cosine
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("fda800"),
    Formula(Equal(ChebyshevT(n, x), Cos(Mul(n, Acos(x))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

2fc479

T_{n}\!\left(x\right) = \cosh\!\left(n \operatorname{acosh}(x)\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = \cosh\!\left(n \operatorname{acosh}(x)\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("2fc479"),
    Formula(Equal(ChebyshevT(n, x), Cosh(Mul(n, Acosh(x))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

b8fdcd

U_{n - 1}\!\left(x\right) \sqrt{1 - {x}^{2}} = \sin\!\left(n \operatorname{acos}(x)\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{n - 1}\!\left(x\right) \sqrt{1 - {x}^{2}} = \sin\!\left(n \operatorname{acos}(x)\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b8fdcd"),
    Formula(Equal(Mul(ChebyshevU(Sub(n, 1), x), Sqrt(Sub(1, Pow(x, 2)))), Sin(Mul(n, Acos(x))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

f4b3fa

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

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

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

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Cos`	$\cos(z)$	Cosine
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f4b3fa"),
    Formula(Equal(ChebyshevT(n, Cos(x)), Cos(Mul(n, x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

4c7aeb

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

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

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

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4c7aeb"),
    Formula(Equal(Mul(ChebyshevU(n, Cos(x)), Sin(x)), Sin(Mul(n, x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

9789ee

T_{2 n + 1}\!\left(\sin(x)\right) = {\left(-1\right)}^{n} \sin\!\left(\left(2 n + 1\right) x\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{2 n + 1}\!\left(\sin(x)\right) = {\left(-1\right)}^{n} \sin\!\left(\left(2 n + 1\right) x\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

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

Source code for this entry:

Entry(ID("9789ee"),
    Formula(Equal(ChebyshevT(Add(Mul(2, n), 1), Sin(x)), Mul(Pow(-1, n), Sin(Mul(Add(Mul(2, n), 1), x))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

0cbe75

T_{n}\!\left(x\right) = \frac{1}{2} \left({\left(x + \sqrt{{x}^{2} - 1}\right)}^{n} + {\left(x - \sqrt{{x}^{2} - 1}\right)}^{n}\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = \frac{1}{2} \left({\left(x + \sqrt{{x}^{2} - 1}\right)}^{n} + {\left(x - \sqrt{{x}^{2} - 1}\right)}^{n}\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; 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
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0cbe75"),
    Formula(Equal(ChebyshevT(n, x), Mul(Div(1, 2), Add(Pow(Add(x, Sqrt(Sub(Pow(x, 2), 1))), n), Pow(Sub(x, Sqrt(Sub(Pow(x, 2), 1))), n))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

61375f

U_{n - 1}\!\left(x\right) \sqrt{{x}^{2} - 1} = \frac{1}{2} \left({\left(x + \sqrt{{x}^{2} - 1}\right)}^{n} - {\left(x - \sqrt{{x}^{2} - 1}\right)}^{n}\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{n - 1}\!\left(x\right) \sqrt{{x}^{2} - 1} = \frac{1}{2} \left({\left(x + \sqrt{{x}^{2} - 1}\right)}^{n} - {\left(x - \sqrt{{x}^{2} - 1}\right)}^{n}\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("61375f"),
    Formula(Equal(Mul(ChebyshevU(Sub(n, 1), x), Sqrt(Sub(Pow(x, 2), 1))), Mul(Div(1, 2), Sub(Pow(Add(x, Sqrt(Sub(Pow(x, 2), 1))), n), Pow(Sub(x, Sqrt(Sub(Pow(x, 2), 1))), n))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

fdf80d

T_{n}\!\left(x\right) + U_{n - 1}\!\left(x\right) \sqrt{{x}^{2} - 1} = {\left(x + \sqrt{{x}^{2} - 1}\right)}^{n}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) + U_{n - 1}\!\left(x\right) \sqrt{{x}^{2} - 1} = {\left(x + \sqrt{{x}^{2} - 1}\right)}^{n}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("fdf80d"),
    Formula(Equal(Add(ChebyshevT(n, x), Mul(ChebyshevU(Sub(n, 1), x), Sqrt(Sub(Pow(x, 2), 1)))), Pow(Add(x, Sqrt(Sub(Pow(x, 2), 1))), n))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

42eb01

T_{n}^{2}\!\left(x\right) + \left({x}^{2} - 1\right) U_{n - 1}^{2}\!\left(x\right) = 1

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}^{2}\!\left(x\right) + \left({x}^{2} - 1\right) U_{n - 1}^{2}\!\left(x\right) = 1

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

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

Source code for this entry:

Entry(ID("42eb01"),
    Formula(Equal(Add(Pow(ChebyshevT(n, x), 2), Mul(Sub(Pow(x, 2), 1), Pow(ChebyshevU(Sub(n, 1), x), 2))), 1)),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

5bd0ec

T_{n}\!\left(\frac{x + {x}^{-1}}{2}\right) = \frac{{x}^{n} + {x}^{-n}}{2}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{0\right\}

TeX:

T_{n}\!\left(\frac{x + {x}^{-1}}{2}\right) = \frac{{x}^{n} + {x}^{-n}}{2}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

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

Source code for this entry:

Entry(ID("5bd0ec"),
    Formula(Equal(ChebyshevT(n, Div(Add(x, Pow(x, -1)), 2)), Div(Add(Pow(x, n), Pow(x, Neg(n))), 2))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, SetMinus(CC, Set(0))))))

305a29

T_{n}\!\left(x\right) = {2}^{n - 1} \prod_{k=1}^{n} \left(x - \cos\!\left(\frac{2 k - 1}{2 n} \pi\right)\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = {2}^{n - 1} \prod_{k=1}^{n} \left(x - \cos\!\left(\frac{2 k - 1}{2 n} \pi\right)\right)

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; 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
`Product`	$\prod_{n} f(n)$	Product
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("305a29"),
    Formula(Equal(ChebyshevT(n, x), Mul(Pow(2, Sub(n, 1)), Product(Parentheses(Sub(x, Cos(Mul(Div(Sub(Mul(2, k), 1), Mul(2, n)), Pi)))), For(k, 1, n))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, CC))))

f5fa23

U_{n}\!\left(x\right) = {2}^{n} \prod_{k=1}^{n} \left(x - \cos\!\left(\frac{k}{n + 1} \pi\right)\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{n}\!\left(x\right) = {2}^{n} \prod_{k=1}^{n} \left(x - \cos\!\left(\frac{k}{n + 1} \pi\right)\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; 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
`Product`	$\prod_{n} f(n)$	Product
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f5fa23"),
    Formula(Equal(ChebyshevU(n, x), Mul(Pow(2, n), Product(Parentheses(Sub(x, Cos(Mul(Div(k, Add(n, 1)), Pi)))), For(k, 1, n))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, CC))))

99aa38

T_{n}\!\left(x\right) = \frac{n}{2} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} \frac{{\left(-1\right)}^{k}}{n - k} {n - k \choose k} {\left(2 x\right)}^{n - 2 k}

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = \frac{n}{2} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} \frac{{\left(-1\right)}^{k}}{n - k} {n - k \choose k} {\left(2 x\right)}^{n - 2 k}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("99aa38"),
    Formula(Equal(ChebyshevT(n, x), Mul(Div(n, 2), Sum(Mul(Mul(Div(Pow(-1, k), Sub(n, k)), Binomial(Sub(n, k), k)), Pow(Mul(2, x), Sub(n, Mul(2, k)))), For(k, 0, Floor(Div(n, 2))))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, CC))))

50cb6b

U_{n}\!\left(x\right) = \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n - k \choose k} {\left(2 x\right)}^{n - 2 k}

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{n}\!\left(x\right) = \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n - k \choose k} {\left(2 x\right)}^{n - 2 k}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("50cb6b"),
    Formula(Equal(ChebyshevU(n, x), Sum(Mul(Mul(Pow(-1, k), Binomial(Sub(n, k), k)), Pow(Mul(2, x), Sub(n, Mul(2, k)))), For(k, 0, Floor(Div(n, 2)))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, CC))))

ae791d

T_{n}\!\left(x\right) = \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {n \choose 2 k} {\left({x}^{2} - 1\right)}^{k} {x}^{n - 2 k}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {n \choose 2 k} {\left({x}^{2} - 1\right)}^{k} {x}^{n - 2 k}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Sum`	$\sum_{n} f(n)$	Sum
`Binomial`	${n \choose k}$	Binomial coefficient
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("ae791d"),
    Formula(Equal(ChebyshevT(n, x), Sum(Mul(Mul(Binomial(n, Mul(2, k)), Pow(Sub(Pow(x, 2), 1), k)), Pow(x, Sub(n, Mul(2, k)))), For(k, 0, Floor(Div(n, 2)))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, CC))))

4f3e30

T_{n}\!\left(x\right) = \frac{n}{2} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} \frac{{\left(-1\right)}^{k} \left(n - k - 1\right)!}{k ! \left(n - 2 k\right)!} {\left(2 x\right)}^{n - 2 k}

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = \frac{n}{2} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} \frac{{\left(-1\right)}^{k} \left(n - k - 1\right)!}{k ! \left(n - 2 k\right)!} {\left(2 x\right)}^{n - 2 k}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4f3e30"),
    Formula(Equal(ChebyshevT(n, x), Mul(Div(n, 2), Sum(Mul(Div(Mul(Pow(-1, k), Factorial(Sub(Sub(n, k), 1))), Mul(Factorial(k), Factorial(Sub(n, Mul(2, k))))), Pow(Mul(2, x), Sub(n, Mul(2, k)))), For(k, 0, Floor(Div(n, 2))))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, CC))))

e9232b

T_{n}\!\left(x\right) = n \sum_{k=0}^{n} \frac{{2}^{k} \left(n + k - 1\right)!}{\left(n - k\right)! \left(2 k\right)!} {\left(x - 1\right)}^{k}

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = n \sum_{k=0}^{n} \frac{{2}^{k} \left(n + k - 1\right)!}{\left(n - k\right)! \left(2 k\right)!} {\left(x - 1\right)}^{k}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("e9232b"),
    Formula(Equal(ChebyshevT(n, x), Mul(n, Sum(Mul(Div(Mul(Pow(2, k), Factorial(Sub(Add(n, k), 1))), Mul(Factorial(Sub(n, k)), Factorial(Mul(2, k)))), Pow(Sub(x, 1), k)), For(k, 0, n))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, CC))))

4e914f

U_{n}\!\left(x\right) = \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {n + 1 \choose 2 k + 1} {\left({x}^{2} - 1\right)}^{k} {x}^{n - 2 k}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{n}\!\left(x\right) = \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {n + 1 \choose 2 k + 1} {\left({x}^{2} - 1\right)}^{k} {x}^{n - 2 k}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Sum`	$\sum_{n} f(n)$	Sum
`Binomial`	${n \choose k}$	Binomial coefficient
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4e914f"),
    Formula(Equal(ChebyshevU(n, x), Sum(Mul(Mul(Binomial(Add(n, 1), Add(Mul(2, k), 1)), Pow(Sub(Pow(x, 2), 1), k)), Pow(x, Sub(n, Mul(2, k)))), For(k, 0, Floor(Div(n, 2)))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, CC))))

a9077a

U_{n}\!\left(x\right) = \sum_{k=0}^{n} \frac{{2}^{k} \left(n + k + 1\right)!}{\left(n - k\right)! \left(2 k + 1\right)!} {\left(x - 1\right)}^{k}

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{n}\!\left(x\right) = \sum_{k=0}^{n} \frac{{2}^{k} \left(n + k + 1\right)!}{\left(n - k\right)! \left(2 k + 1\right)!} {\left(x - 1\right)}^{k}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a9077a"),
    Formula(Equal(ChebyshevU(n, x), Sum(Mul(Div(Mul(Pow(2, k), Factorial(Add(Add(n, k), 1))), Mul(Factorial(Sub(n, k)), Factorial(Add(Mul(2, k), 1)))), Pow(Sub(x, 1), k)), For(k, 0, n)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, CC))))

382679

T_{n}\!\left(x\right) = \,{}_2F_1\!\left(-n, n, \frac{1}{2}, \frac{1 - x}{2}\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = \,{}_2F_1\!\left(-n, n, \frac{1}{2}, \frac{1 - x}{2}\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("382679"),
    Formula(Equal(ChebyshevT(n, x), Hypergeometric2F1(Neg(n), n, Div(1, 2), Div(Sub(1, x), 2)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

ce9a39

U_{n}\!\left(x\right) = \left(n + 1\right) \,{}_2F_1\!\left(-n, n + 2, \frac{3}{2}, \frac{1 - x}{2}\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{n}\!\left(x\right) = \left(n + 1\right) \,{}_2F_1\!\left(-n, n + 2, \frac{3}{2}, \frac{1 - x}{2}\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("ce9a39"),
    Formula(Equal(ChebyshevU(n, x), Mul(Add(n, 1), Hypergeometric2F1(Neg(n), Add(n, 2), Div(3, 2), Div(Sub(1, x), 2))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

685d1a

\sum_{n=0}^{\infty} T_{n}\!\left(x\right) {z}^{n} = \frac{1 - x z}{1 - 2 x z + {z}^{2}}

Assumptions:

x \in \left[-1, 1\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

TeX:

\sum_{n=0}^{\infty} T_{n}\!\left(x\right) {z}^{n} = \frac{1 - x z}{1 - 2 x z + {z}^{2}}

x \in \left[-1, 1\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("685d1a"),
    Formula(Equal(Sum(Mul(ChebyshevT(n, x), Pow(z, n)), For(n, 0, Infinity)), Div(Sub(1, Mul(x, z)), Add(Sub(1, Mul(Mul(2, x), z)), Pow(z, 2))))),
    Variables(x, z),
    Assumptions(And(Element(x, ClosedInterval(-1, 1)), Element(z, CC), Less(Abs(z), 1))))

b5049d

\sum_{n=0}^{\infty} U_{n}\!\left(x\right) {z}^{n} = \frac{1}{1 - 2 x z + {z}^{2}}

Assumptions:

x \in \left[-1, 1\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

TeX:

\sum_{n=0}^{\infty} U_{n}\!\left(x\right) {z}^{n} = \frac{1}{1 - 2 x z + {z}^{2}}

x \in \left[-1, 1\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("b5049d"),
    Formula(Equal(Sum(Mul(ChebyshevU(n, x), Pow(z, n)), For(n, 0, Infinity)), Div(1, Add(Sub(1, Mul(Mul(2, x), z)), Pow(z, 2))))),
    Variables(x, z),
    Assumptions(And(Element(x, ClosedInterval(-1, 1)), Element(z, CC), Less(Abs(z), 1))))

27b2bb

\sum_{n=1}^{\infty} T_{n}\!\left(x\right) \frac{{z}^{n}}{n} = -\frac{1}{2} \log\!\left(1 - 2 x z + {z}^{2}\right)

Assumptions:

x \in \left[-1, 1\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

TeX:

\sum_{n=1}^{\infty} T_{n}\!\left(x\right) \frac{{z}^{n}}{n} = -\frac{1}{2} \log\!\left(1 - 2 x z + {z}^{2}\right)

x \in \left[-1, 1\right] \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Log`	$\log(z)$	Natural logarithm
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("27b2bb"),
    Formula(Equal(Sum(Mul(ChebyshevT(n, x), Div(Pow(z, n), n)), For(n, 1, Infinity)), Mul(Neg(Div(1, 2)), Log(Add(Sub(1, Mul(Mul(2, x), z)), Pow(z, 2)))))),
    Variables(x, z),
    Assumptions(And(Element(x, ClosedInterval(-1, 1)), Element(z, CC), Less(Abs(z), 1))))

9d7c61

\sum_{n=0}^{\infty} T_{n}\!\left(x\right) \frac{{z}^{n}}{n !} = {e}^{z x} \cosh\!\left(z \sqrt{{x}^{2} - 1}\right)

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\sum_{n=0}^{\infty} T_{n}\!\left(x\right) \frac{{z}^{n}}{n !} = {e}^{z x} \cosh\!\left(z \sqrt{{x}^{2} - 1}\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Sqrt`	$\sqrt{z}$	Principal square root
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9d7c61"),
    Formula(Equal(Sum(Mul(ChebyshevT(n, x), Div(Pow(z, n), Factorial(n))), For(n, 0, Infinity)), Mul(Exp(Mul(z, x)), Cosh(Mul(z, Sqrt(Sub(Pow(x, 2), 1))))))),
    Variables(x, z),
    Assumptions(And(Element(x, CC), Element(z, CC))))

fff8ff

\sum_{n=0}^{\infty} U_{n}\!\left(x\right) \frac{{z}^{n}}{n !} = {e}^{z x} \left(\cosh\!\left(z \sqrt{{x}^{2} - 1}\right) + z x \operatorname{sinc}\!\left(i z \sqrt{{x}^{2} - 1}\right)\right)

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\sum_{n=0}^{\infty} U_{n}\!\left(x\right) \frac{{z}^{n}}{n !} = {e}^{z x} \left(\cosh\!\left(z \sqrt{{x}^{2} - 1}\right) + z x \operatorname{sinc}\!\left(i z \sqrt{{x}^{2} - 1}\right)\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`Sqrt`	$\sqrt{z}$	Principal square root
`Sinc`	$\operatorname{sinc}(z)$	Sinc function
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("fff8ff"),
    Formula(Equal(Sum(Mul(ChebyshevU(n, x), Div(Pow(z, n), Factorial(n))), For(n, 0, Infinity)), Mul(Exp(Mul(z, x)), Add(Cosh(Mul(z, Sqrt(Sub(Pow(x, 2), 1)))), Mul(Mul(z, x), Sinc(Mul(Mul(ConstI, z), Sqrt(Sub(Pow(x, 2), 1))))))))),
    Variables(x, z),
    Assumptions(And(Element(x, CC), Element(z, CC))))

1a0d11

T'_{n}(x) = n U_{n - 1}\!\left(x\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T'_{n}(x) = n U_{n - 1}\!\left(x\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("1a0d11"),
    Formula(Equal(ComplexDerivative(ChebyshevT(n, x), For(x, x)), Mul(n, ChebyshevU(Sub(n, 1), x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

05fe07

T''_{n}(x) = \frac{n \left(n T_{n}\!\left(x\right) - x U_{n - 1}\!\left(x\right)\right)}{{x}^{2} - 1}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}

TeX:

T''_{n}(x) = \frac{n \left(n T_{n}\!\left(x\right) - x U_{n - 1}\!\left(x\right)\right)}{{x}^{2} - 1}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("05fe07"),
    Formula(Equal(ComplexDerivative(ChebyshevT(n, x), For(x, x, 2)), Div(Mul(n, Sub(Mul(n, ChebyshevT(n, x)), Mul(x, ChebyshevU(Sub(n, 1), x)))), Sub(Pow(x, 2), 1)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, SetMinus(CC, Set(-1, 1))))))

35e13b

U'_{n}(x) = \frac{\left(n + 1\right) T_{n + 1}\!\left(x\right) - x U_{n}\!\left(x\right)}{{x}^{2} - 1}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}

TeX:

U'_{n}(x) = \frac{\left(n + 1\right) T_{n + 1}\!\left(x\right) - x U_{n}\!\left(x\right)}{{x}^{2} - 1}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("35e13b"),
    Formula(Equal(ComplexDerivative(ChebyshevU(n, x), For(x, x)), Div(Sub(Mul(Add(n, 1), ChebyshevT(Add(n, 1), x)), Mul(x, ChebyshevU(n, x))), Sub(Pow(x, 2), 1)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, SetMinus(CC, Set(-1, 1))))))

12ce84

{T}^{(r)}_{n}(x) = \frac{\left(n\right)_{r} \left(n - r + 1\right)_{r}}{\left(2 r - 1\right)!!} \,{}_2F_1\!\left(r + n, r - n, \frac{1}{2} + r, \frac{1 - x}{2}\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(r \le n \;\mathbin{\operatorname{or}}\; x \ne -1\right)

TeX:

{T}^{(r)}_{n}(x) = \frac{\left(n\right)_{r} \left(n - r + 1\right)_{r}}{\left(2 r - 1\right)!!} \,{}_2F_1\!\left(r + n, r - n, \frac{1}{2} + r, \frac{1 - x}{2}\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(r \le n \;\mathbin{\operatorname{or}}\; x \ne -1\right)

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("12ce84"),
    Formula(Equal(ComplexDerivative(ChebyshevT(n, x), For(x, x, r)), Mul(Div(Mul(RisingFactorial(n, r), RisingFactorial(Add(Sub(n, r), 1), r)), DoubleFactorial(Sub(Mul(2, r), 1))), Hypergeometric2F1(Add(r, n), Sub(r, n), Add(Div(1, 2), r), Div(Sub(1, x), 2))))),
    Variables(n, r, x),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)), Element(x, CC), Or(LessEqual(r, n), NotEqual(x, -1)))))

9d66de

{U}^{(r)}_{n}(x) = \frac{\left(n + 1\right)_{r + 1} \left(n - r + 1\right)_{r}}{\left(2 r + 1\right)!!} \,{}_2F_1\!\left(r + n + 2, r - n, \frac{3}{2} + r, \frac{1 - x}{2}\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(r \le n \;\mathbin{\operatorname{or}}\; x \ne -1\right)

TeX:

{U}^{(r)}_{n}(x) = \frac{\left(n + 1\right)_{r + 1} \left(n - r + 1\right)_{r}}{\left(2 r + 1\right)!!} \,{}_2F_1\!\left(r + n + 2, r - n, \frac{3}{2} + r, \frac{1 - x}{2}\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(r \le n \;\mathbin{\operatorname{or}}\; x \ne -1\right)

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9d66de"),
    Formula(Equal(ComplexDerivative(ChebyshevU(n, x), For(x, x, r)), Mul(Div(Mul(RisingFactorial(Add(n, 1), Add(r, 1)), RisingFactorial(Add(Sub(n, r), 1), r)), DoubleFactorial(Add(Mul(2, r), 1))), Hypergeometric2F1(Add(Add(r, n), 2), Sub(r, n), Add(Div(3, 2), r), Div(Sub(1, x), 2))))),
    Variables(n, r, x),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)), Element(x, CC), Or(LessEqual(r, n), NotEqual(x, -1)))))

a68f0e

{T}^{(r)}_{n}(1) = \frac{\left(n\right)_{r} \left(n - r + 1\right)_{r}}{\left(2 r - 1\right)!!}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

{T}^{(r)}_{n}(1) = \frac{\left(n\right)_{r} \left(n - r + 1\right)_{r}}{\left(2 r - 1\right)!!}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("a68f0e"),
    Formula(Equal(ComplexDerivative(ChebyshevT(n, x), For(x, 1, r)), Div(Mul(RisingFactorial(n, r), RisingFactorial(Add(Sub(n, r), 1), r)), DoubleFactorial(Sub(Mul(2, r), 1))))),
    Variables(n, r),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)))))

b6b014

{U}^{(r)}_{n}(1) = \frac{\left(n + 1\right)_{r + 1} \left(n - r + 1\right)_{r}}{\left(2 r + 1\right)!!}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

{U}^{(r)}_{n}(1) = \frac{\left(n + 1\right)_{r + 1} \left(n - r + 1\right)_{r}}{\left(2 r + 1\right)!!}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("b6b014"),
    Formula(Equal(ComplexDerivative(ChebyshevU(n, x), For(x, 1, r)), Div(Mul(RisingFactorial(Add(n, 1), Add(r, 1)), RisingFactorial(Add(Sub(n, r), 1), r)), DoubleFactorial(Add(Mul(2, r), 1))))),
    Variables(n, r),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)))))

6582c4

{T}^{(r)}_{n}(x) = \frac{\sqrt{\pi}}{{\left(x - 1\right)}^{r}} \,{}_3{\textbf F}_2\!\left(1, -n, n, \frac{1}{2}, 1 - r, \frac{1 - x}{2}\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}

References:

http://functions.wolfram.com/Polynomials/ChebyshevT/20/02/01/0002/

TeX:

{T}^{(r)}_{n}(x) = \frac{\sqrt{\pi}}{{\left(x - 1\right)}^{r}} \,{}_3{\textbf F}_2\!\left(1, -n, n, \frac{1}{2}, 1 - r, \frac{1 - x}{2}\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("6582c4"),
    Formula(Equal(ComplexDerivative(ChebyshevT(n, x), For(x, x, r)), Mul(Div(Sqrt(Pi), Pow(Sub(x, 1), r)), Hypergeometric3F2Regularized(1, Neg(n), n, Div(1, 2), Sub(1, r), Div(Sub(1, x), 2))))),
    Variables(n, r, x),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)), Element(x, SetMinus(CC, Set(-1, 1))))),
    References("http://functions.wolfram.com/Polynomials/ChebyshevT/20/02/01/0002/"))

e1797b

{U}^{(r)}_{n}(x) = \frac{\sqrt{\pi} \left(n + 1\right)}{2 {\left(x - 1\right)}^{r}} \,{}_3{\textbf F}_2\!\left(1, -n, n + 2, \frac{3}{2}, 1 - r, \frac{1 - x}{2}\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}

References:

http://functions.wolfram.com/Polynomials/ChebyshevU/20/02/01/0002/

TeX:

{U}^{(r)}_{n}(x) = \frac{\sqrt{\pi} \left(n + 1\right)}{2 {\left(x - 1\right)}^{r}} \,{}_3{\textbf F}_2\!\left(1, -n, n + 2, \frac{3}{2}, 1 - r, \frac{1 - x}{2}\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("e1797b"),
    Formula(Equal(ComplexDerivative(ChebyshevU(n, x), For(x, x, r)), Mul(Div(Mul(Sqrt(Pi), Add(n, 1)), Mul(2, Pow(Sub(x, 1), r))), Hypergeometric3F2Regularized(1, Neg(n), Add(n, 2), Div(3, 2), Sub(1, r), Div(Sub(1, x), 2))))),
    Variables(n, r, x),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)), Element(x, SetMinus(CC, Set(-1, 1))))),
    References("http://functions.wolfram.com/Polynomials/ChebyshevU/20/02/01/0002/"))

15dd69

\left|T_{n}\!\left(x\right)\right| \le 1

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \left[-1, 1\right]

TeX:

\left|T_{n}\!\left(x\right)\right| \le 1

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \left[-1, 1\right]

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`ClosedInterval`	$\left[a, b\right]$	Closed interval

Source code for this entry:

Entry(ID("15dd69"),
    Formula(LessEqual(Abs(ChebyshevT(n, x)), 1)),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, ClosedInterval(-1, 1)))))

3c662e

\left|U_{n}\!\left(x\right)\right| \le \left|n + 1\right|

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \left[-1, 1\right]

TeX:

\left|U_{n}\!\left(x\right)\right| \le \left|n + 1\right|

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \left[-1, 1\right]

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ZZ`	$\mathbb{Z}$	Integers
`ClosedInterval`	$\left[a, b\right]$	Closed interval

Source code for this entry:

Entry(ID("3c662e"),
    Formula(LessEqual(Abs(ChebyshevU(n, x)), Abs(Add(n, 1)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, ClosedInterval(-1, 1)))))

c718ea

\left|T_{n}\!\left(z\right)\right| \le \left|T_{n}\!\left(i \left|z\right|\right)\right|

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\left|T_{n}\!\left(z\right)\right| \le \left|T_{n}\!\left(i \left|z\right|\right)\right|

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ConstI`	$i$	Imaginary unit
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("c718ea"),
    Formula(LessEqual(Abs(ChebyshevT(n, z)), Abs(ChebyshevT(n, Mul(ConstI, Abs(z)))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZ), Element(z, CC))))

0b3fd6

\left|U_{n}\!\left(z\right)\right| \le \left|U_{n}\!\left(i \left|z\right|\right)\right|

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\left|U_{n}\!\left(z\right)\right| \le \left|U_{n}\!\left(i \left|z\right|\right)\right|

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ConstI`	$i$	Imaginary unit
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0b3fd6"),
    Formula(LessEqual(Abs(ChebyshevU(n, z)), Abs(ChebyshevU(n, Mul(ConstI, Abs(z)))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZ), Element(z, CC))))

443759

\left|T_{n}\!\left(z\right)\right| \le {\left(\left|z\right| + \sqrt{{\left|z\right|}^{2} + 1}\right)}^{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\left|T_{n}\!\left(z\right)\right| \le {\left(\left|z\right| + \sqrt{{\left|z\right|}^{2} + 1}\right)}^{n}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("443759"),
    Formula(LessEqual(Abs(ChebyshevT(n, z)), Pow(Add(Abs(z), Sqrt(Add(Pow(Abs(z), 2), 1))), n))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

2a4b9d

\left|U_{n}\!\left(z\right)\right| \le {\left(\left|z\right| + \sqrt{{\left|z\right|}^{2} + 1}\right)}^{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\left|U_{n}\!\left(z\right)\right| \le {\left(\left|z\right| + \sqrt{{\left|z\right|}^{2} + 1}\right)}^{n}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("2a4b9d"),
    Formula(LessEqual(Abs(ChebyshevU(n, z)), Pow(Add(Abs(z), Sqrt(Add(Pow(Abs(z), 2), 1))), n))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

b0c84b

{\left(T_{n}\!\left(x\right)\right)}^{2} - T_{n - 1}\!\left(x\right) T_{n + 1}\!\left(x\right) \ge 0

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \left[-1, 1\right]

TeX:

{\left(T_{n}\!\left(x\right)\right)}^{2} - T_{n - 1}\!\left(x\right) T_{n + 1}\!\left(x\right) \ge 0

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \left[-1, 1\right]

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ClosedInterval`	$\left[a, b\right]$	Closed interval

Source code for this entry:

Entry(ID("b0c84b"),
    Formula(GreaterEqual(Sub(Pow(Parentheses(ChebyshevT(n, x)), 2), Mul(ChebyshevT(Sub(n, 1), x), ChebyshevT(Add(n, 1), x))), 0)),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, ClosedInterval(-1, 1)))))

2ada0f

{\left(T_{n}\!\left(x\right)\right)}^{2} - T_{n - 1}\!\left(x\right) T_{n + 1}\!\left(x\right) > 0

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \left(-1, 1\right)

TeX:

{\left(T_{n}\!\left(x\right)\right)}^{2} - T_{n - 1}\!\left(x\right) T_{n + 1}\!\left(x\right) > 0

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \left(-1, 1\right)

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("2ada0f"),
    Formula(Greater(Sub(Pow(Parentheses(ChebyshevT(n, x)), 2), Mul(ChebyshevT(Sub(n, 1), x), ChebyshevT(Add(n, 1), x))), 0)),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, OpenInterval(-1, 1)))))

54be3e

\left|{T}^{(r)}_{n - 1}(x)\right| \le \left|{T}^{(r)}_{n}(x)\right|

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left|x\right| \ge 1

References:

V. V. Prasolov, Polynomials, Springer, 2004, ISBN 978-3-642-03980-5. Theorem 3.14.10.

TeX:

\left|{T}^{(r)}_{n - 1}(x)\right| \le \left|{T}^{(r)}_{n}(x)\right|

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left|x\right| \ge 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("54be3e"),
    Formula(LessEqual(Abs(ComplexDerivative(ChebyshevT(Sub(n, 1), x), For(x, x, r))), Abs(ComplexDerivative(ChebyshevT(n, x), For(x, x, r))))),
    Variables(n, r, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, RR), GreaterEqual(Abs(x), 1))),
    References("V. V. Prasolov, Polynomials, Springer, 2004, ISBN 978-3-642-03980-5. Theorem 3.14.10."))

f61927

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

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \left[1, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[1, \infty\right)

References:

V. V. Prasolov, Polynomials, Springer, 2004, ISBN 978-3-642-03980-5. Theorem 3.14.11.
R. Askey, An inequality for Tchebycheff polynomials and extensions, Journal of Approximation Theory, 1975, Volume 14, pp 1-11

TeX:

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

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \left[1, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("f61927"),
    Formula(LessEqual(ChebyshevT(n, Mul(x, y)), Mul(ChebyshevT(n, x), ChebyshevT(n, y)))),
    Variables(n, x, y),
    Assumptions(And(Element(n, ZZ), Element(x, ClosedOpenInterval(1, Infinity)), Element(y, ClosedOpenInterval(1, Infinity)))),
    References("V. V. Prasolov, Polynomials, Springer, 2004, ISBN 978-3-642-03980-5. Theorem 3.14.11.", "R. Askey, An inequality for Tchebycheff polynomials and extensions, Journal of Approximation Theory, 1975, Volume 14, pp 1-11"))

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Derivatives

Bounds and inequalities

Upper bounds

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