ChebyshevU

The Chebyshev polynomial of the second kind $U_n(x)$, defined by $U_n(\cos\theta)\sin\theta = \sin((n+1)\theta)$. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaChebyshev polynomials↗MathWorldChebyshevPolynomialoftheSecondKind↗DLMF18.3↗FungrimChebyshevU✓↗Fungrim entryWolfram LanguageChebyshevU!↗

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

Typed signatures

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

Domain: Special functions

Details

• $U_0 = 1$, $U_1 = 2x$, and $U_{n+1} = 2x\,U_n - U_{n-1}$: the recurrence of ChebyshevT from a different start. The coefficients are integers and the leading one is $2^n$.
• $U_n(\cos\theta) = \dfrac{\sin((n+1)\theta)}{\sin\theta}$, so $U_n(1) = n + 1$ and $U_n(-1) = (-1)^n (n+1)$.
• With the first kind: $T_n = \tfrac12(U_n - U_{n-2})$, and $T_n(x)^2 - (x^2-1)\,U_{n-1}(x)^2 = 1$.
• A negative order folds as Wolfram does: $U_{-1} = 0$ and $U_{-n} = -U_{n-2}$ for $n \ge 2$.
• 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 $U_\nu$ to non-integer $\nu$, and that is not implemented here.

Examples

7 more kept as data, checked against the oracles

Basic #

$U_2(x) = 4x^2 - 1$

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

$U_3(x) = 8x^3 - 4x$

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

At a floating-point $x$

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In$\mathrm{ChebyshevU}(5, 0.7)$Out$-1.397\,76$

Evaluates at an integer: $U_2(3) = 4\cdot9 - 1$

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In$\mathrm{ChebyshevU}(2, 3)$Out$35$

The degree-10 polynomial in $x$

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In$\mathrm{ChebyshevU}(10, x)$Out$1\,024x^{10}-2\,304x^8+1\,792x^6-560x^4+60x^2-1$

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

Scope #

At a rational $x$ the value is exact

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In$\mathrm{ChebyshevU}(3, \frac{1}{2})$Out$-1$

At a Gaussian integer

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In$\mathrm{ChebyshevU}(4, 2+3i)$Out$-1\,843-2\,064i$

A symbolic order stays unevaluated

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

Exact integer argument

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In$\mathrm{ChebyshevU}(2, 5)$Out$99$

$U_0$ is constant 1

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

$U_0$ is 1 even at infinity

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In$\mathrm{ChebyshevU}(0, ∞)$Out$1$

Symbolic argument gives the odd degree-7 polynomial

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In$\mathrm{ChebyshevU}(7, x)$Out$128x^7-192x^5+80x^3-8x$

Odd degree: $U_1(-x) = -U_1(x)$

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

Even degree: $U_2(-x) = U_2(x)$

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

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

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In$\mathrm{MatrixFunction}(()↦\mathrm{ChebyshevU}(2, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{1}{66}(\sqrt{33}(33^{1/2}-3)((5+33^{1/2})^2-1))-\frac{1}{66}(\sqrt{33}(-33^{1/2}-3)((5-33^{1/2})^2-1)), \frac{1}{33}(2\sqrt{33}(5+33^{1/2})^2)-\frac{1}{33}(2\sqrt{33}(5-33^{1/2})^2)\bigr\rbrack, \bigl\lbrack\frac{1}{11}(\sqrt{33}(5+33^{1/2})^2)-\frac{1}{11}(\sqrt{33}(5-33^{1/2})^2), \frac{1}{66}(\sqrt{33}((5+33^{1/2})^2-1)(3+\sqrt{33}))-\frac{1}{66}(\sqrt{33}((5-33^{1/2})^2-1)(3-33^{1/2}))\bigr\rbrack\bigr\rbrack$

Properties #

$U_n(1) = n + 1$

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In$\mathrm{ChebyshevU}(5, 1)$Out$6$

$U_n(-1) = (-1)^n (n + 1)$

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In$\mathrm{ChebyshevU}(5, -1)$Out$-6$

$U_{n+1} = 2x\,U_n - U_{n-1}$, here at $x = \tfrac13$

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In$\mathrm{ChebyshevU}(4, \frac{1}{3})-(\frac{1}{3}(2\mathrm{ChebyshevU}(3, 1/3))-\mathrm{ChebyshevU}(2, \frac{1}{3}))$Out$0$

$U_{-1} = 0$

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In$\mathrm{ChebyshevU}(-1, x)$Out$0$

Below that a negative order folds: $U_{-n} = -U_{n-2}$

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In$\mathrm{ChebyshevU}(-5, x)$Out$4x-8x^3$

$U_0(1) = 1$

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

Known value · Wolfram MathematicalFunctionData, ChebyshevU

$U_0$ is 1 for a complex argument too

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In$\mathrm{ChebyshevU}(0, -\frac{1}{2}i)$Out$1$

Known value · Wolfram MathematicalFunctionData, ChebyshevU

Complex argument: $U_1(x) = 2x$ gives $-i$

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In$\mathrm{ChebyshevU}(1, -\frac{1}{2}i)$Out$-i$

Known value · Wolfram MathematicalFunctionData, ChebyshevU

$U_1$ at infinity grows without bound

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In$\mathrm{ChebyshevU}(1, ∞)$Out$∞$

Known value · Wolfram MathematicalFunctionData, ChebyshevU:Polynomial

Likewise toward negative infinity

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In$\mathrm{ChebyshevU}(1, -∞)$Out$-∞$

Known value · Wolfram MathematicalFunctionData, ChebyshevU:Polynomial

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In$\mathrm{ChebyshevU}(2, ∞)$Out$∞$

Known value · Wolfram MathematicalFunctionData, ChebyshevU:Polynomial

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In$\mathrm{ChebyshevU}(2, -∞)$Out$∞$

Known value · Wolfram MathematicalFunctionData, ChebyshevU:Polynomial

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In$\sum_{k=0}^2\mathrm{ChebyshevU}(k, \frac{1}{2})\mathrm{ChebyshevU}(k, 2)$Out$5$

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

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In$\sum_{k=0}^3\mathrm{ChebyshevU}(k, \frac{1}{2})\mathrm{ChebyshevU}(k, 2)$Out$-51$

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

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In$N(\int_{-1}^{1}\!\sqrt{1-t^2}\mathrm{ChebyshevU}(2, t)^2\, \mathrm{d}t)$Out$1.570796326794973±7.638334409421077e-14$

Known value · Wolfram MathematicalFunctionData, ChebyshevU:Polynomial, Orthogonality relation

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In$N(\int_{-1}^{1}\!\mathrm{ChebyshevU}(2, t)\mathrm{ChebyshevU}(3, t)\sqrt{1-t^2}\, \mathrm{d}t)$Out$0±9.468215858034646e-13$

Known value · Wolfram MathematicalFunctionData, ChebyshevU:Polynomial, Orthogonality relation

About

Examples	39
Checked against	Wolfram Language 37 agree2 differ
Crosswalk	WikipediaMathWorldDLMFFungrimFungrim entry ×30Wolfram 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

ChebyshevU[n, x]; mpmath.chebyu(n, x); sympy.chebyshevu(n, x).

See also: ChebyshevT, LegendrePolynomial, Sin