Orthogonal polynomials

Part of the Fungrim Identities reference — 74 identities for orthogonal polynomials.

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Chebyshev polynomials (51)
Gaussian quadrature (1)
Legendre polynomials (22)

Chebyshev polynomials

$x\mapsto\mathrm{ChebyshevT}(n, x)^{\doubleprime}(x)=\frac{n(n\mathrm{ChebyshevT}(n, x)-x\mathrm{ChebyshevU}(n-1, x))}{x^2-1}$

Holds when $n\in\Z\land x\in\C\setminus\lbrace-1, 1\rbrace$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind; ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. 05fe07 · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, x)=\frac{1}{2}(\mathrm{ChebyshevU}(n, x)-\mathrm{ChebyshevU}(n-2, x))$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind; ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. 0649c9 · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, x)=\frac{1}{2}({(x-(x^2-1)^{1/2})}^{n}+{(x+\sqrt{x^2-1})}^{n})$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for expansion. 0cbe75 · Fungrim entry ↗

$x\mapsto\mathrm{ChebyshevT}(n, x)^{\prime}(x)=n\mathrm{ChebyshevU}(n-1, x)$

$\mathrm{ChebyshevT}(n, -1)=(-1)^{n}$

Holds when $n\in\Z$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. 2760e7 · Fungrim entry ↗

$\mathrm{ChebyshevU}(2n, 0)=(-1)^{n}$

Holds when $n\in\Z$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. 2a5337 · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, x)=\cosh(n\mathrm{arcosh}(x))$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. 2fc479 · Fungrim entry ↗

$\mathrm{ChebyshevU}(n, x)=2x\mathrm{ChebyshevU}(n+1, x)-\mathrm{ChebyshevU}(n+2, x)$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. 303204 · Fungrim entry ↗

$x\mapsto\mathrm{ChebyshevU}(n, x)^{\prime}(x)=\frac{(n+1)\mathrm{ChebyshevT}(n+1, x)-x\mathrm{ChebyshevU}(n, x)}{x^2-1}$

$\mathrm{ChebyshevT}(n, x)=\mathrm{Hypergeometric2F_1}(-n, n, \frac{1}{2}, \frac{1-x}{2})$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind; Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 382679 · Fungrim entry ↗

$\mathrm{ChebyshevT}(2n+1, 0)=0$

Holds when $n\in\Z$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. 42102c · Fungrim entry ↗

$(x^2-1)\mathrm{ChebyshevU}(n-1, x)^2+\mathrm{ChebyshevT}(n, x)^2=1$

$\mathrm{ChebyshevU}(0, x)=1$

Holds when $x\in\C$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. 48765b · Fungrim entry ↗

$\mathrm{ChebyshevT}(2n, x)=2\mathrm{ChebyshevT}(n, x)^2-1$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. 4b83c6 · Fungrim entry ↗

$\sin(x)\mathrm{ChebyshevU}(n, \cos(x))=\sin(nx)$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. 4c7aeb · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, \frac{x+\frac{1}{x}}{2})=\frac{1}{2}(x^{n}+x^{-n})$

Holds when $n\in\Z\land x\in\C\setminus\lbrace0\rbrace$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for expansion. 5bd0ec · Fungrim entry ↗

$\mathrm{ChebyshevU}(2n, x)=\mathrm{ChebyshevU}(n-1, 2x^2-1)+\mathrm{ChebyshevT}(n, 2x^2-1)$

$\mathrm{ChebyshevU}(n-1, x)\sqrt{x^2-1}=\frac{1}{2}({(x+(x^2-1)^{1/2})}^{n}-{(x-(x^2-1)^{1/2})}^{n})$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. 61375f · Fungrim entry ↗

$x\mapsto\mathrm{ChebyshevT}(n, x)^{\prime}(x)=\frac{\mathrm{Hypergeometric3F2Regularized}(1, -n, n, \frac{1}{2}, 1-r, \frac{1-x}{2})\sqrt{\pi}}{(x-1)^{r}}$

Holds when $n\in\Z\land r\in\N\land x\in\C\setminus\lbrace-1, 1\rbrace$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for expansion. Reference: functions.wolfram.com 6582c4 · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, -x)=\mathrm{ChebyshevT}(n, x)\times(-1)^{n}$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. 6a24ab · Fungrim entry ↗

$\mathrm{ChebyshevU}(1, x)=2x$

Holds when $x\in\C$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. 75eacb · Fungrim entry ↗

$\mathrm{ChebyshevU}(-n, x)=-\mathrm{ChebyshevU}(n-2, x)$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. 78f5bb · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, x)=x\mathrm{ChebyshevT}(n-1, x)-(1-x^2)\mathrm{ChebyshevU}(n-2, x)$

$\mathrm{ChebyshevU}(2n+1, 0)=0$

Holds when $n\in\Z$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. 7d111e · Fungrim entry ↗

$\mathrm{ChebyshevT}(m, \mathrm{ChebyshevT}(n, x))=\mathrm{ChebyshevT}(mn, x)$

Holds when $m\in\Z\land n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. 7e882c · Fungrim entry ↗

$\mathrm{ChebyshevT}(2n, x)=\mathrm{ChebyshevT}(n, 2x^2-1)$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. 82288c · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, x)=\mathrm{ChebyshevU}(n, x)-x\mathrm{ChebyshevU}(n-1, x)$

$\mathrm{ChebyshevU}(n, -x)=\mathrm{ChebyshevU}(n, x)\times(-1)^{n}$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. 88aeb6 · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, x)=2x\mathrm{ChebyshevT}(n+1, x)-\mathrm{ChebyshevT}(n+2, x)$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. 8a785a · Fungrim entry ↗

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

Holds when $x\in\C$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. 9001e6 · Fungrim entry ↗

$\mathrm{ChebyshevT}(-n, x)=\mathrm{ChebyshevT}(n, x)$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. 9093a3 · Fungrim entry ↗

$\mathrm{ChebyshevT}(2n+1, \sin(x))=\sin(x(2n+1))\times(-1)^{n}$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. 9789ee · Fungrim entry ↗

$\mathrm{ChebyshevT}(2n, 0)=(-1)^{n}$

Holds when $n\in\Z$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. a46d91 · Fungrim entry ↗

$x\mapsto\mathrm{ChebyshevT}(n, x)^{\prime}(1)=\frac{\mathrm{RisingFactorial}(n, r)\mathrm{RisingFactorial}(n-r+1, r)}{(2r-1)!!}$

Holds when $n\in\Z\land r\in\N$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind; RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. a68f0e · Fungrim entry ↗

$x\mapsto\mathrm{ChebyshevU}(n, x)^{\prime}(1)=\frac{\mathrm{RisingFactorial}(n+1, r+1)\mathrm{RisingFactorial}(n-r+1, r)}{(2r+1)!!}$

Holds when $n\in\Z\land r\in\N$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind; RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. b6b014 · Fungrim entry ↗

$\mathrm{ChebyshevU}(n-1, x)\sqrt{1-x^2}=\sin(n\arccos(x))$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. b8fdcd · Fungrim entry ↗

$\mathrm{ChebyshevT}(1, x)=x$

Holds when $x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. be5652 · Fungrim entry ↗

$\mathrm{ChebyshevU}(n, -1)=(n+1)\times(-1)^{n}$

Holds when $n\in\Z$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for expansion. be9a45 · Fungrim entry ↗

$\mathrm{ChebyshevT}(0, x)=1$

Holds when $x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. c76e72 · Fungrim entry ↗

$\mathrm{ChebyshevU}(n, x)=x\mathrm{ChebyshevU}(n-1, x)+\mathrm{ChebyshevT}(n, x)$

$\mathrm{ChebyshevU}(n, x)=(n+1)\mathrm{Hypergeometric2F_1}(-n, n+2, \frac{3}{2}, \frac{1-x}{2})$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind; Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. ce9a39 · Fungrim entry ↗

$\mathrm{ChebyshevU}(n, x)=2x\mathrm{ChebyshevU}(n-1, x)-\mathrm{ChebyshevU}(n-2, x)$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. d1ef91 · Fungrim entry ↗

$\mathrm{ChebyshevT}(2n+1, x)=2\mathrm{ChebyshevT}(n, x)\mathrm{ChebyshevT}(n+1, x)-x$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. de0968 · Fungrim entry ↗

$\mathrm{ChebyshevU}(n, 1)=n+1$

Holds when $n\in\Z$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. e03fa4 · Fungrim entry ↗

$x\mapsto\mathrm{ChebyshevU}(n, x)^{\prime}(x)=\frac{(n+1)\mathrm{Hypergeometric3F2Regularized}(1, -n, n+2, \frac{3}{2}, 1-r, \frac{1-x}{2})\sqrt{\pi}}{2(x-1)^{r}}$

Holds when $n\in\Z\land r\in\N\land x\in\C\setminus\lbrace-1, 1\rbrace$ . Symbols: ChebyshevU — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. Reference: functions.wolfram.com e1797b · Fungrim entry ↗

$\mathrm{ChebyshevT}(m, x)\mathrm{ChebyshevT}(n, x)=\frac{1}{2}(\mathrm{ChebyshevT}(m+n, x)+\mathrm{ChebyshevT}(\vert m-n\vert, x))$

Holds when $m\in\Z\land n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. ed5222 · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, \cos(x))=\cos(nx)$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. f4b3fa · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, x)=2x\mathrm{ChebyshevT}(n-1, x)-\mathrm{ChebyshevT}(n-2, x)$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. faeed9 · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, 1)=1$

Holds when $n\in\Z$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. fc5d42 · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, x)=\cos(n\arccos(x))$

Holds when $n\in\Z\land x\in\C$ . Symbols: ChebyshevT — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. fda800 · Fungrim entry ↗

$\mathrm{ChebyshevT}(n, x)+\mathrm{ChebyshevU}(n-1, x)\sqrt{x^2-1}={(x+\sqrt{x^2-1})}^{n}$

Gaussian quadrature

$\mathrm{GaussLegendreWeight}(n, k)=(2)((1-\mathrm{LegendrePolynomialZero}(n, k)^2)t\mapsto\mathrm{LegendrePolynomial}(n, t)^{\prime}(\mathrm{LegendrePolynomialZero}(n, k))^2)^{-1}$

Holds when $n\in\N^*\land k\in1..n$ . Used by the Compute Engine for simplification. ea4754 · Fungrim entry ↗

Legendre polynomials

$\mathrm{LegendrePolynomial}(n, -z)=\mathrm{LegendrePolynomial}(n, z)\times(-1)^{n}$

Holds when $n\in\N\land z\in\C$ . Used by the Compute Engine for simplification. 0010f3 · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(5, z)=\frac{1}{8}(63z^5-70z^3+15z)$

Holds when $z\in\C$ . Used by the Compute Engine for simplification. 13f971 · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(1, z)=z$

Holds when $z\in\C$ . Used by the Compute Engine for simplification. 217521 · Fungrim entry ↗

$(1-z^2)z\mapsto\mathrm{LegendrePolynomial}(n, z)^{\doubleprime}(z)+n(n+1)\mathrm{LegendrePolynomial}(n, z)-2zz\mapsto\mathrm{LegendrePolynomial}(n, z)^{\prime}(z)=0$

Holds when $n\in\N\land z\in\C$ . Used by the Compute Engine for simplification. 27688e · Fungrim entry ↗

$-(z(2n+1)\mathrm{LegendrePolynomial}(n, z))+n\mathrm{LegendrePolynomial}(n-1, z)+(n+1)\mathrm{LegendrePolynomial}(n+1, z)=0$

Holds when $n\in\N^*\land z\in\C$ . Used by the Compute Engine for simplification. 367ac2 · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(n, z)=\mathrm{Hypergeometric2F_1}(-n, -n, 1, \frac{z+1}{z-1})(\frac{z-1}{2})^{n}$

Holds when $n\in\N\land z\in\C\setminus\lbrace1\rbrace$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 3c87b9 · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(n, -1)=(-1)^{n}$

Holds when $n\in\N$ . Used by the Compute Engine for simplification. 3df748 · Fungrim entry ↗

$\mathrm{Count}(\mathrm{Zeros}(z\mapsto\mathrm{LegendrePolynomial}(n, z), \C))=n$

Holds when $n\in\N$ . Symbols: Zeros — Zeros (roots) of function. Used by the Compute Engine for simplification. 415911 · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(n, z)=\frac{t\mapsto{(t^2-1)}^{n}^{\prime}(z)}{n!\times2^{n}}$

Holds when $n\in\N$ or $z\in\C$ . Used by the Compute Engine for simplification. 4cfeac · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(2n, 0)=\frac{\mathrm{Binomial}(2n, n)\times(-1)^{n}}{4^{n}}$

Holds when $n\in\N$ . Used by the Compute Engine for simplification. 674afa · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(2n, z)=\frac{\mathrm{Hypergeometric2F_1}(-n, n+\frac{1}{2}, \frac{1}{2}, z^2)\mathrm{Binomial}(2n, n)\times(-1)^{n}}{4^{n}}$

Holds when $n\in\N\land z\in\C$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 6cd4a1 · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(2n+1, z)=\frac{z(2n+1)\mathrm{Hypergeometric2F_1}(-n, n+\frac{3}{2}, \frac{3}{2}, z^2)\mathrm{Binomial}(2n, n)\times(-1)^{n}}{4^{n}}$

Holds when $n\in\N\land z\in\C$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 859445 · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(2n+1, 0)=0$

Holds when $n\in\N$ . Used by the Compute Engine for simplification. 85eebc · Fungrim entry ↗

$(1-z^2)z\mapsto\mathrm{LegendrePolynomial}(n, z)^{\prime}(z)+nz\mathrm{LegendrePolynomial}(n, z)-n\mathrm{LegendrePolynomial}(n-1, z)=0$

Holds when $n\in\N^*\land z\in\C$ . Used by the Compute Engine for simplification. 925fdf · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(n, z)=\mathrm{Hypergeometric2F_1}(-n, n+1, 1, \frac{1-z}{2})$

Holds when $n\in\N\land z\in\C$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 9395fc · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(3, z)=\frac{1}{2}(5z^3-3z)$

Holds when $z\in\C$ . Used by the Compute Engine for expansion. 9b7f05 · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(0, z)=1$

Holds when $z\in\C$ . Used by the Compute Engine for simplification. 9bdf22 · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(4, z)=\frac{1}{8}(35z^4-30z^2+3)$

Holds when $z\in\C$ . Used by the Compute Engine for simplification. a17386 · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(n, 1)=1$

Holds when $n\in\N$ . Used by the Compute Engine for simplification. a7ac51 · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(n, z^\star)=\mathrm{LegendrePolynomial}(n, z)^\star$

Holds when $n\in\N\land z\in\C$ . Used by the Compute Engine for expansion. b2d723 · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(2, z)=\frac{1}{2}(3z^2-1)$

Holds when $z\in\C$ . Used by the Compute Engine for expansion. d77f0a · Fungrim entry ↗

$\mathrm{LegendrePolynomial}(n, z)=\mathrm{Hypergeometric2F_1}(-(\frac{n}{2}), \frac{1-n}{2}, \frac{1}{2}-n, \frac{1}{z^2})\mathrm{Binomial}(2n, n)(\frac{z}{2})^{n}$

Holds when $n\in\N\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. f55f0a · Fungrim entry ↗

Contents​

Chebyshev polynomials​

Gaussian quadrature​

Legendre polynomials​

Contents

Chebyshev polynomials

Gaussian quadrature

Legendre polynomials