# Orthogonal polynomials Part of the [Fungrim Identities](/compute-engine/reference/fungrim/) reference — **74 identities** for orthogonal polynomials. :::info[Generated reference] This page is generated from the compiled Fungrim artifact by `scripts/fungrim/gen-reference-doc.ts` (upstream snapshot `3a299164c683`, translator `grim2mathjson 0.1.0`). Do not edit it by hand. The corpus is MIT-licensed; see `data/fungrim/LICENSE`. ::: ## Contents - [Chebyshev polynomials](#chebyshev-polynomials) (51) - [Gaussian quadrature](#gaussian-quadrature) (1) - [Legendre polynomials](#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 ↗](https://fungrim.org/entry/05fe07) --- $$\mathrm{ChebyshevT}(n, x)=\frac{1}{2}(\mathrm{ChebyshevU}(n, x)-\mathrm{ChebyshevU}(n-2, x))$$ **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 ↗](https://fungrim.org/entry/0649c9) --- $$\mathrm{ChebyshevT}(n, x)=\frac{1}{2}({(x+\sqrt{x^2-1})}^{n}+{(x-(x^2-1)^{1/2})}^{n})$$ **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 ↗](https://fungrim.org/entry/0cbe75) --- $$x\mapsto\mathrm{ChebyshevT}(n, x)^{\prime}(x)=n\mathrm{ChebyshevU}(n-1, x)$$ **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. [`1a0d11` · Fungrim entry ↗](https://fungrim.org/entry/1a0d11) --- $$\mathrm{ChebyshevT}(n, -1)=(-1)^{n}$$ **Holds when** $n\in\Z$. **Symbols:** **ChebyshevT** — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. [`2760e7` · Fungrim entry ↗](https://fungrim.org/entry/2760e7) --- $$\mathrm{ChebyshevU}(2n, 0)=(-1)^{n}$$ **Holds when** $n\in\Z$. **Symbols:** **ChebyshevU** — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. [`2a5337` · Fungrim entry ↗](https://fungrim.org/entry/2a5337) --- $$\mathrm{ChebyshevT}(n, x)=\cosh(n\mathrm{arcosh}(x))$$ **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 ↗](https://fungrim.org/entry/2fc479) --- $$\mathrm{ChebyshevU}(n, x)=2x\mathrm{ChebyshevU}(n+1, x)-\mathrm{ChebyshevU}(n+2, x)$$ **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 ↗](https://fungrim.org/entry/303204) --- $$x\mapsto\mathrm{ChebyshevU}(n, x)^{\prime}(x)=\frac{(n+1)\mathrm{ChebyshevT}(n+1, x)-x\mathrm{ChebyshevU}(n, x)}{x^2-1}$$ **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. [`35e13b` · Fungrim entry ↗](https://fungrim.org/entry/35e13b) --- $$\mathrm{ChebyshevT}(n, x)=\mathrm{Hypergeometric2F_1}(-n, n, \frac{1}{2}, \frac{1-x}{2})$$ **Holds when** $n\in\Z\land x\in\C$. **Symbols:** **ChebyshevT** — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. [`382679` · Fungrim entry ↗](https://fungrim.org/entry/382679) --- $$\mathrm{ChebyshevT}(2n+1, 0)=0$$ **Holds when** $n\in\Z$. **Symbols:** **ChebyshevT** — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. [`42102c` · Fungrim entry ↗](https://fungrim.org/entry/42102c) --- $$\mathrm{ChebyshevT}(n, x)^2+(x^2-1)\mathrm{ChebyshevU}(n-1, x)^2=1$$ **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. [`42eb01` · Fungrim entry ↗](https://fungrim.org/entry/42eb01) --- $$\mathrm{ChebyshevU}(0, x)=1$$ **Holds when** $x\in\C$. **Symbols:** **ChebyshevU** — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. [`48765b` · Fungrim entry ↗](https://fungrim.org/entry/48765b) --- $$\mathrm{ChebyshevT}(2n, x)=2\mathrm{ChebyshevT}(n, x)^2-1$$ **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 ↗](https://fungrim.org/entry/4b83c6) --- $$\mathrm{ChebyshevU}(n, \cos(x))\sin(x)=\sin(nx)$$ **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 ↗](https://fungrim.org/entry/4c7aeb) --- $$\mathrm{ChebyshevT}(n, \frac{x+\frac{1}{x}}{2})=\frac{1}{2}(x^{n}+x^{-n})$$ **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 ↗](https://fungrim.org/entry/5bd0ec) --- $$\mathrm{ChebyshevU}(2n, x)=\mathrm{ChebyshevT}(n, 2x^2-1)+\mathrm{ChebyshevU}(n-1, 2x^2-1)$$ **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. [`5f09f4` · Fungrim entry ↗](https://fungrim.org/entry/5f09f4) --- $$\mathrm{ChebyshevU}(n-1, x)\sqrt{x^2-1}=\frac{1}{2}({(x+(x^2-1)^{1/2})}^{n}-{(x-(x^2-1)^{1/2})}^{n})$$ **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 ↗](https://fungrim.org/entry/61375f) --- $$x\mapsto\mathrm{ChebyshevT}(n, x)^{\prime}(x)=\frac{\mathrm{Hypergeometric3F2Regularized}(1, -n, n, \frac{1}{2}, 1-r, \frac{1-x}{2})\sqrt{\pi}}{(x-1)^{r}}$$ **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](http://functions.wolfram.com/Polynomials/ChebyshevT/20/02/01/0002/) [`6582c4` · Fungrim entry ↗](https://fungrim.org/entry/6582c4) --- $$\mathrm{ChebyshevT}(n, -x)=(-1)^{n}\mathrm{ChebyshevT}(n, x)$$ **Holds when** $n\in\Z\land x\in\C$. **Symbols:** **ChebyshevT** — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. [`6a24ab` · Fungrim entry ↗](https://fungrim.org/entry/6a24ab) --- $$\mathrm{ChebyshevU}(1, x)=2x$$ **Holds when** $x\in\C$. **Symbols:** **ChebyshevU** — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. [`75eacb` · Fungrim entry ↗](https://fungrim.org/entry/75eacb) --- $$\mathrm{ChebyshevU}(-n, x)=-\mathrm{ChebyshevU}(n-2, x)$$ **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 ↗](https://fungrim.org/entry/78f5bb) --- $$\mathrm{ChebyshevT}(n, x)=x\mathrm{ChebyshevT}(n-1, x)-(1-x^2)\mathrm{ChebyshevU}(n-2, x)$$ **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. [`7b2c26` · Fungrim entry ↗](https://fungrim.org/entry/7b2c26) --- $$\mathrm{ChebyshevU}(2n+1, 0)=0$$ **Holds when** $n\in\Z$. **Symbols:** **ChebyshevU** — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. [`7d111e` · Fungrim entry ↗](https://fungrim.org/entry/7d111e) --- $$\mathrm{ChebyshevT}(m, \mathrm{ChebyshevT}(n, x))=\mathrm{ChebyshevT}(mn, x)$$ **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 ↗](https://fungrim.org/entry/7e882c) --- $$\mathrm{ChebyshevT}(2n, x)=\mathrm{ChebyshevT}(n, 2x^2-1)$$ **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 ↗](https://fungrim.org/entry/82288c) --- $$\mathrm{ChebyshevT}(n, x)=\mathrm{ChebyshevU}(n, x)-x\mathrm{ChebyshevU}(n-1, x)$$ **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. [`844561` · Fungrim entry ↗](https://fungrim.org/entry/844561) --- $$\mathrm{ChebyshevU}(n, -x)=(-1)^{n}\mathrm{ChebyshevU}(n, x)$$ **Holds when** $n\in\Z\land x\in\C$. **Symbols:** **ChebyshevU** — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. [`88aeb6` · Fungrim entry ↗](https://fungrim.org/entry/88aeb6) --- $$\mathrm{ChebyshevT}(n, x)=2x\mathrm{ChebyshevT}(n+1, x)-\mathrm{ChebyshevT}(n+2, x)$$ **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 ↗](https://fungrim.org/entry/8a785a) --- $$\mathrm{ChebyshevU}(-1, x)=0$$ **Holds when** $x\in\C$. **Symbols:** **ChebyshevU** — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. [`9001e6` · Fungrim entry ↗](https://fungrim.org/entry/9001e6) --- $$\mathrm{ChebyshevT}(-n, x)=\mathrm{ChebyshevT}(n, x)$$ **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 ↗](https://fungrim.org/entry/9093a3) --- $$\mathrm{ChebyshevT}(2n+1, \sin(x))=(-1)^{n}\sin((2n+1)x)$$ **Holds when** $n\in\Z\land x\in\C$. **Symbols:** **ChebyshevT** — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. [`9789ee` · Fungrim entry ↗](https://fungrim.org/entry/9789ee) --- $$\mathrm{ChebyshevT}(2n, 0)=(-1)^{n}$$ **Holds when** $n\in\Z$. **Symbols:** **ChebyshevT** — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. [`a46d91` · Fungrim entry ↗](https://fungrim.org/entry/a46d91) --- $$x\mapsto\mathrm{ChebyshevT}(n, x)^{\prime}(1)=\frac{\mathrm{RisingFactorial}(n, r)\mathrm{RisingFactorial}(n-r+1, r)}{(2r-1)!!}$$ **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 ↗](https://fungrim.org/entry/a68f0e) --- $$x\mapsto\mathrm{ChebyshevU}(n, x)^{\prime}(1)=\frac{\mathrm{RisingFactorial}(n+1, r+1)\mathrm{RisingFactorial}(n-r+1, r)}{(2r+1)!!}$$ **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 ↗](https://fungrim.org/entry/b6b014) --- $$\mathrm{ChebyshevU}(n-1, x)\sqrt{1-x^2}=\sin(n\arccos(x))$$ **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 ↗](https://fungrim.org/entry/b8fdcd) --- $$\mathrm{ChebyshevT}(1, x)=x$$ **Holds when** $x\in\C$. **Symbols:** **ChebyshevT** — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. [`be5652` · Fungrim entry ↗](https://fungrim.org/entry/be5652) --- $$\mathrm{ChebyshevU}(n, -1)=(-1)^{n}(n+1)$$ **Holds when** $n\in\Z$. **Symbols:** **ChebyshevU** — Chebyshev polynomial of the second kind. Used by the Compute Engine for expansion. [`be9a45` · Fungrim entry ↗](https://fungrim.org/entry/be9a45) --- $$\mathrm{ChebyshevT}(0, x)=1$$ **Holds when** $x\in\C$. **Symbols:** **ChebyshevT** — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. [`c76e72` · Fungrim entry ↗](https://fungrim.org/entry/c76e72) --- $$\mathrm{ChebyshevU}(n, x)=x\mathrm{ChebyshevU}(n-1, x)+\mathrm{ChebyshevT}(n, x)$$ **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. [`ce5e03` · Fungrim entry ↗](https://fungrim.org/entry/ce5e03) --- $$\mathrm{ChebyshevU}(n, x)=(n+1)\mathrm{Hypergeometric2F_1}(-n, n+2, \frac{3}{2}, \frac{1-x}{2})$$ **Holds when** $n\in\Z\land x\in\C$. **Symbols:** **ChebyshevU** — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. [`ce9a39` · Fungrim entry ↗](https://fungrim.org/entry/ce9a39) --- $$\mathrm{ChebyshevU}(n, x)=2x\mathrm{ChebyshevU}(n-1, x)-\mathrm{ChebyshevU}(n-2, x)$$ **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 ↗](https://fungrim.org/entry/d1ef91) --- $$\mathrm{ChebyshevT}(2n+1, x)=2\mathrm{ChebyshevT}(n+1, x)\mathrm{ChebyshevT}(n, x)-x$$ **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 ↗](https://fungrim.org/entry/de0968) --- $$\mathrm{ChebyshevU}(n, 1)=n+1$$ **Holds when** $n\in\Z$. **Symbols:** **ChebyshevU** — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. [`e03fa4` · Fungrim entry ↗](https://fungrim.org/entry/e03fa4) --- $$x\mapsto\mathrm{ChebyshevU}(n, x)^{\prime}(x)=\frac{(n+1)\mathrm{Hypergeometric3F2Regularized}(1, -n, n+2, \frac{3}{2}, 1-r, \frac{1-x}{2})\sqrt{\pi}}{2(x-1)^{r}}$$ **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](http://functions.wolfram.com/Polynomials/ChebyshevU/20/02/01/0002/) [`e1797b` · Fungrim entry ↗](https://fungrim.org/entry/e1797b) --- $$\mathrm{ChebyshevT}(m, x)\mathrm{ChebyshevT}(n, x)=\frac{1}{2}(\mathrm{ChebyshevT}(m+n, x)+\mathrm{ChebyshevT}(\vert m-n\vert, x))$$ **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 ↗](https://fungrim.org/entry/ed5222) --- $$\mathrm{ChebyshevT}(n, \cos(x))=\cos(nx)$$ **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 ↗](https://fungrim.org/entry/f4b3fa) --- $$\mathrm{ChebyshevT}(n, x)=2x\mathrm{ChebyshevT}(n-1, x)-\mathrm{ChebyshevT}(n-2, x)$$ **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 ↗](https://fungrim.org/entry/faeed9) --- $$\mathrm{ChebyshevT}(n, 1)=1$$ **Holds when** $n\in\Z$. **Symbols:** **ChebyshevT** — Chebyshev polynomial of the first kind. Used by the Compute Engine for simplification. [`fc5d42` · Fungrim entry ↗](https://fungrim.org/entry/fc5d42) --- $$\mathrm{ChebyshevT}(n, x)=\cos(n\arccos(x))$$ **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 ↗](https://fungrim.org/entry/fda800) --- $$\mathrm{ChebyshevT}(n, x)+\mathrm{ChebyshevU}(n-1, x)\sqrt{x^2-1}={(x+\sqrt{x^2-1})}^{n}$$ **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. [`fdf80d` · Fungrim entry ↗](https://fungrim.org/entry/fdf80d) --- ## 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 ↗](https://fungrim.org/entry/ea4754) --- ## Legendre polynomials $$\mathrm{LegendrePolynomial}(n, -z)=(-1)^{n}\mathrm{LegendrePolynomial}(n, z)$$ **Holds when** $n\in\N\land z\in\C$. Used by the Compute Engine for simplification. [`0010f3` · Fungrim entry ↗](https://fungrim.org/entry/0010f3) --- $$\mathrm{LegendrePolynomial}(5, z)=\frac{1}{8}(63z^5-70z^3+15z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`13f971` · Fungrim entry ↗](https://fungrim.org/entry/13f971) --- $$\mathrm{LegendrePolynomial}(1, z)=z$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`217521` · Fungrim entry ↗](https://fungrim.org/entry/217521) --- $$(1-z^2)z\mapsto\mathrm{LegendrePolynomial}(n, z)^{\doubleprime}(z)-2zz\mapsto\mathrm{LegendrePolynomial}(n, z)^{\prime}(z)+n(n+1)\mathrm{LegendrePolynomial}(n, z)=0$$ **Holds when** $n\in\N\land z\in\C$. Used by the Compute Engine for simplification. [`27688e` · Fungrim entry ↗](https://fungrim.org/entry/27688e) --- $$(n+1)\mathrm{LegendrePolynomial}(n+1, z)-(2n+1)z\mathrm{LegendrePolynomial}(n, z)+n\mathrm{LegendrePolynomial}(n-1, z)=0$$ **Holds when** $n\in\N^*\land z\in\C$. Used by the Compute Engine for simplification. [`367ac2` · Fungrim entry ↗](https://fungrim.org/entry/367ac2) --- $$\mathrm{LegendrePolynomial}(n, z)=\frac{z-1}{2}^{n}\mathrm{Hypergeometric2F_1}(-n, -n, 1, \frac{z+1}{z-1})$$ **Holds when** $n\in\N\land z\in\C\setminus\lbrace1\rbrace$. Used by the Compute Engine for simplification. [`3c87b9` · Fungrim entry ↗](https://fungrim.org/entry/3c87b9) --- $$\mathrm{LegendrePolynomial}(n, -1)=(-1)^{n}$$ **Holds when** $n\in\N$. Used by the Compute Engine for simplification. [`3df748` · Fungrim entry ↗](https://fungrim.org/entry/3df748) --- $$\mathrm{Count}(\mathrm{Zeros}(z\mapsto\mathrm{LegendrePolynomial}(n, z), \C))=n$$ **Holds when** $n\in\N$. **Symbols:** **Zeros** — Zeros (roots) of function. Used by the Compute Engine for simplification. [`415911` · Fungrim entry ↗](https://fungrim.org/entry/415911) --- $$\mathrm{LegendrePolynomial}(n, z)=\frac{t\mapsto{(t^2-1)}^{n}^{\prime}(z)}{n!\times2^{n}}$$ **Holds when** $n\in\N$  _or_  $z\in\C$. Used by the Compute Engine for simplification. [`4cfeac` · Fungrim entry ↗](https://fungrim.org/entry/4cfeac) --- $$\mathrm{LegendrePolynomial}(2n, 0)=\frac{\binom{2n}{n}\times(-1)^{n}}{4^{n}}$$ **Holds when** $n\in\N$. Used by the Compute Engine for simplification. [`674afa` · Fungrim entry ↗](https://fungrim.org/entry/674afa) --- $$\mathrm{LegendrePolynomial}(2n, z)=\frac{\mathrm{Hypergeometric2F_1}(-n, n+\frac{1}{2}, \frac{1}{2}, z^2)\binom{2n}{n}\times(-1)^{n}}{4^{n}}$$ **Holds when** $n\in\N\land z\in\C$. Used by the Compute Engine for simplification. [`6cd4a1` · Fungrim entry ↗](https://fungrim.org/entry/6cd4a1) --- $$\mathrm{LegendrePolynomial}(2n+1, z)=\frac{z(2n+1)\mathrm{Hypergeometric2F_1}(-n, n+\frac{3}{2}, \frac{3}{2}, z^2)\binom{2n}{n}\times(-1)^{n}}{4^{n}}$$ **Holds when** $n\in\N\land z\in\C$. Used by the Compute Engine for simplification. [`859445` · Fungrim entry ↗](https://fungrim.org/entry/859445) --- $$\mathrm{LegendrePolynomial}(2n+1, 0)=0$$ **Holds when** $n\in\N$. Used by the Compute Engine for simplification. [`85eebc` · Fungrim entry ↗](https://fungrim.org/entry/85eebc) --- $$((1-z^2)z\mapsto\mathrm{LegendrePolynomial}(n, z)^{\prime}(z)+nz\mathrm{LegendrePolynomial}(n, z))-n\mathrm{LegendrePolynomial}(n-1, z)=0$$ **Holds when** $n\in\N^*\land z\in\C$. Used by the Compute Engine for simplification. [`925fdf` · Fungrim entry ↗](https://fungrim.org/entry/925fdf) --- $$\mathrm{LegendrePolynomial}(n, z)=\mathrm{Hypergeometric2F_1}(-n, n+1, 1, \frac{1-z}{2})$$ **Holds when** $n\in\N\land z\in\C$. Used by the Compute Engine for simplification. [`9395fc` · Fungrim entry ↗](https://fungrim.org/entry/9395fc) --- $$\mathrm{LegendrePolynomial}(3, z)=\frac{1}{2}(5z^3-3z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`9b7f05` · Fungrim entry ↗](https://fungrim.org/entry/9b7f05) --- $$\mathrm{LegendrePolynomial}(0, z)=1$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`9bdf22` · Fungrim entry ↗](https://fungrim.org/entry/9bdf22) --- $$\mathrm{LegendrePolynomial}(4, z)=\frac{1}{8}(35z^4-30z^2+3)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`a17386` · Fungrim entry ↗](https://fungrim.org/entry/a17386) --- $$\mathrm{LegendrePolynomial}(n, 1)=1$$ **Holds when** $n\in\N$. Used by the Compute Engine for simplification. [`a7ac51` · Fungrim entry ↗](https://fungrim.org/entry/a7ac51) --- $$\mathrm{LegendrePolynomial}(n, z^\star)=\mathrm{LegendrePolynomial}(n, z)^\star$$ **Holds when** $n\in\N\land z\in\C$. Used by the Compute Engine for expansion. [`b2d723` · Fungrim entry ↗](https://fungrim.org/entry/b2d723) --- $$\mathrm{LegendrePolynomial}(2, z)=\frac{1}{2}(3z^2-1)$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`d77f0a` · Fungrim entry ↗](https://fungrim.org/entry/d77f0a) --- $$\mathrm{LegendrePolynomial}(n, z)=\binom{2n}{n}\frac{z}{2}^{n}\mathrm{Hypergeometric2F_1}(-(\frac{n}{2}), \frac{1-n}{2}, \frac{1}{2}-n, \frac{1}{z^2})$$ **Holds when** $n\in\N\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`f55f0a` · Fungrim entry ↗](https://fungrim.org/entry/f55f0a) ---