ChebyshevT

The Chebyshev polynomial of the first kind $T_n(x)$, defined by $T_n(\cos\theta) = \cos(n\theta)$. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaChebyshev polynomials↗MathWorldChebyshevPolynomialoftheFirstKind↗DLMF18.3↗FungrimChebyshevT✓↗Fungrim entryWolfram LanguageChebyshevT!↗

ChebyshevT(n, x)the degree-$n$ Chebyshev polynomial of the first kind at $x$, for an integer $n$.

Typed signatures

Type	Package	Applies
(integer, number) -> number	analytic	always

Domain: Special functions

Details

• $T_0 = 1$, $T_1 = x$, and $T_{n+1} = 2x\,T_n - T_{n-1}$. The coefficients are integers and the leading one is $2^{n-1}$ for $n \ge 1$.
• $T_n(\cos\theta) = \cos(n\theta)$, so on $[-1, 1]$ the polynomial oscillates between $-1$ and $1$, and $T_n(1) = 1$, $T_n(-1) = (-1)^n$.
• Composition multiplies degrees: $T_m(T_n(x)) = T_{mn}(x)$. Parity follows the degree: $T_n(-x) = (-1)^n T_n(x)$.
• A negative order folds as Wolfram does: $T_{-n} = T_n$.
• At an integer $n$ and a symbolic or exact $x$ the result is the expanded polynomial. At a floating-point or complex $x$ the recurrence runs directly on the number.
• A symbolic order stays unevaluated. The signature takes an integer order only; Wolfram continues $T_\nu$ to non-integer $\nu$, and that is not implemented here.

Examples

7 more kept as data, checked against the oracles

Basic #

$T_3(x) = 4x^3 - 3x$

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In$\mathrm{ChebyshevT}(3, x)$Out$4x^3-3x$

$T_4(x) = 8x^4 - 8x^2 + 1$

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In$\mathrm{ChebyshevT}(4, x)$Out$8x^4-8x^2+1$

At an integer the value is exact

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In$\mathrm{ChebyshevT}(5, 2)$Out$362$

At a floating-point $x$

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In$\mathrm{ChebyshevT}(5, 0.7)$Out$-0.670\,879\,999\,999\,999\,8$

Evaluates at an integer: $T_2(3) = 2\cdot9 - 1$

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In$\mathrm{ChebyshevT}(2, 3)$Out$17$

The degree-10 polynomial in $x$

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In$\mathrm{ChebyshevT}(10, x)$Out$512x^{10}-1\,280x^8+1\,120x^6-400x^4+50x^2-1$

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In$\operatorname{Series}(\mathrm{ChebyshevT}(5, x), x, 0, 5)$Out$x\mathrm{ChebyshevT}^{(0, 1)}(5, 0)+\frac{1}{2}(\mathrm{ChebyshevT}^{(0, 2)}(5, 0)x^2)+\frac{1}{6}(\mathrm{ChebyshevT}^{(0, 3)}(5, 0)x^3)+\frac{1}{24}(\mathrm{ChebyshevT}^{(0, 4)}(5, 0)x^4)+\frac{1}{120}(\mathrm{ChebyshevT}^{(0, 5)}(5, 0)x^5)+O\left(x^6\right)$

Scope #

At a rational $x$ the value is an exact rational

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In$\mathrm{ChebyshevT}(5, \frac{1}{3})$Out$\frac{241}{243}$

At a Gaussian integer

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In$\mathrm{ChebyshevT}(4, 2+3i)$Out$-911-1\,056i$

A symbolic order stays unevaluated

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In$\mathrm{ChebyshevT}(n, x)$Out$\mathrm{ChebyshevT}(n, x)$

Exact integer argument

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In$\mathrm{ChebyshevT}(2, 5)$Out$49$

$T_0$ is constant 1

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In$\mathrm{ChebyshevT}(0, 0)$Out$1$

$T_0$ is 1 even at infinity

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In$\mathrm{ChebyshevT}(0, ∞)$Out$1$

Symbolic argument gives the odd degree-7 polynomial

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In$\mathrm{ChebyshevT}(7, x)$Out$64x^7-112x^5+56x^3-7x$

Odd degree: $T_1(-x) = -T_1(x)$

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In$\mathrm{ChebyshevT}(1, -x)=-\mathrm{ChebyshevT}(1, x)$Out$\mathrm{True}$

Even degree: $T_2(-x) = T_2(x)$

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In$\mathrm{ChebyshevT}(2, -x)=\mathrm{ChebyshevT}(2, x)$Out$\mathrm{True}$

Applied to a matrix as a matrix function, giving exact radicals

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In$\mathrm{MatrixFunction}(()↦\mathrm{ChebyshevT}(2, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{1}{132}(\sqrt{33}(33^{1/2}-3)((5+33^{1/2})^2-2))-\frac{1}{132}(\sqrt{33}(-33^{1/2}-3)((5-33^{1/2})^2-2)), \frac{1}{33}(\sqrt{33}(5+33^{1/2})^2)-\frac{1}{33}(\sqrt{33}(5-33^{1/2})^2)\bigr\rbrack, \bigl\lbrack\frac{1}{22}(\sqrt{33}(5+33^{1/2})^2)-\frac{1}{22}(\sqrt{33}(5-33^{1/2})^2), \frac{1}{132}(\sqrt{33}((5+33^{1/2})^2-2)(3+\sqrt{33}))-\frac{1}{132}(\sqrt{33}((5-33^{1/2})^2-2)(3-33^{1/2}))\bigr\rbrack\bigr\rbrack$

Properties #

$T_n(\cos\theta) = \cos(n\theta)$, here at $\theta = \pi/3$

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In$\mathrm{ChebyshevT}(6, \frac{1}{2})=\cos(\frac{6π}{3})$Out$\mathrm{True}$

$T_n(1) = 1$

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In$\mathrm{ChebyshevT}(7, 1)$Out$1$

$T_n(-1) = (-1)^n$

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In$\mathrm{ChebyshevT}(7, -1)$Out$-1$

Odd degree, odd function: $T_n(-x) = (-1)^n T_n(x)$

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In$\mathrm{ChebyshevT}(5, -\frac{1}{3})=-\mathrm{ChebyshevT}(5, \frac{1}{3})$Out$\mathrm{True}$

A negative order folds: $T_{-n} = T_n$

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In$\mathrm{ChebyshevT}(-3, x)$Out$4x^3-3x$

$T_n(x)^2 - (x^2 - 1)\,U_{n-1}(x)^2 = 1$, here at $x = 3$. See ChebyshevU

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In$\mathrm{ChebyshevT}(4, 3)^2-8\mathrm{ChebyshevU}(3, 3)^2$Out$1$

$T_0(1) = 1$

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In$\mathrm{ChebyshevT}(0, 1)$Out$1$

Known value · Wolfram MathematicalFunctionData, ChebyshevT

$T_1$ at infinity grows without bound

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In$\mathrm{ChebyshevT}(1, ∞)$Out$∞$

Known value · Wolfram MathematicalFunctionData, ChebyshevT:Polynomial

Likewise toward negative infinity

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In$\mathrm{ChebyshevT}(1, -∞)$Out$-∞$

Known value · Wolfram MathematicalFunctionData, ChebyshevT:Polynomial

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In$\mathrm{ChebyshevT}(2, ∞)$Out$∞$

Known value · Wolfram MathematicalFunctionData, ChebyshevT:Polynomial

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In$\mathrm{ChebyshevT}(2, -∞)$Out$∞$

Known value · Wolfram MathematicalFunctionData, ChebyshevT:Polynomial

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In$\sum_{k=0}^2\mathrm{ChebyshevT}(k, \frac{1}{2})\mathrm{ChebyshevT}(k, 2)$Out$-\frac{3}{2}$

Known value · Wolfram MathematicalFunctionData, ChebyshevT:Polynomial, Christoffel‐Darboux formula

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In$\sum_{k=0}^3\mathrm{ChebyshevT}(k, \frac{1}{2})\mathrm{ChebyshevT}(k, 2)$Out$-\frac{55}{2}$

Known value · Wolfram MathematicalFunctionData, ChebyshevT:Polynomial, Christoffel‐Darboux formula

About

Examples	37
Checked against	Wolfram Language 35 agree2 differ
Crosswalk	WikipediaMathWorldDLMFFungrimFungrim entry ×34Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/chebyshev.ts

mappedexternalwolfram / mpmath / sympypackages/oracle/src/mappings.ts

ChebyshevT[n, x]; mpmath.chebyt(n, x); sympy.chebyshevt(n, x).

See also: ChebyshevU, LegendrePolynomial, Cos