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Legendre polynomials

Table of contents: Particular values - Recurrence and functional equations - Generating functions - Rodrigues' formula - Integrals - Sum representations - Hypergeometric representations - Bounds and inequalities - Analytic properties - Gauss-Legendre quadrature - Bounds and inequalities

Particular values

9bdf22
P0 ⁣(z)=1P_{0}\!\left(z\right) = 1
217521
P1 ⁣(z)=zP_{1}\!\left(z\right) = z
d77f0a
P2 ⁣(z)=12(3z21)P_{2}\!\left(z\right) = \frac{1}{2} \left(3 {z}^{2} - 1\right)
9b7f05
P3 ⁣(z)=12(5z33z)P_{3}\!\left(z\right) = \frac{1}{2} \left(5 {z}^{3} - 3 z\right)
a17386
P4 ⁣(z)=18(35z430z2+3)P_{4}\!\left(z\right) = \frac{1}{8} \left(35 {z}^{4} - 30 {z}^{2} + 3\right)
13f971
P5 ⁣(z)=18(63z570z3+15z)P_{5}\!\left(z\right) = \frac{1}{8} \left(63 {z}^{5} - 70 {z}^{3} + 15 z\right)
a7ac51
Pn ⁣(1)=1P_{n}\!\left(1\right) = 1
3df748
Pn ⁣(1)=(1)nP_{n}\!\left(-1\right) = {\left(-1\right)}^{n}
674afa
P2n ⁣(0)=(1)n4n(2nn)P_{2 n}\!\left(0\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} {2 n \choose n}
85eebc
P2n+1 ⁣(0)=0P_{2 n + 1}\!\left(0\right) = 0

Recurrence and functional equations

0010f3
Pn ⁣(z)=(1)nPn ⁣(z)P_{n}\!\left(-z\right) = {\left(-1\right)}^{n} P_{n}\!\left(z\right)
367ac2
(n+1)Pn+1 ⁣(z)(2n+1)zPn ⁣(z)+nPn1 ⁣(z)=0\left(n + 1\right) P_{n + 1}\!\left(z\right) - \left(2 n + 1\right) z P_{n}\!\left(z\right) + n P_{n - 1}\!\left(z\right) = 0
27688e
(1z2)Pn(z)2zPn(z)+n(n+1)Pn ⁣(z)=0\left(1 - {z}^{2}\right) P''_{n}(z) - 2 z P'_{n}(z) + n \left(n + 1\right) P_{n}\!\left(z\right) = 0
925fdf
(1z2)Pn(z)+nzPn ⁣(z)nPn1 ⁣(z)=0\left(1 - {z}^{2}\right) P'_{n}(z) + n z P_{n}\!\left(z\right) - n P_{n - 1}\!\left(z\right) = 0

Generating functions

d84519
n=0Pn ⁣(x)zn=112xz+z2\sum_{n=0}^{\infty} P_{n}\!\left(x\right) {z}^{n} = \frac{1}{\sqrt{1 - 2 x z + {z}^{2}}}

Rodrigues' formula

4cfeac
Pn ⁣(z)=12nn![dndtn(t21)n]t=zP_{n}\!\left(z\right) = \frac{1}{{2}^{n} n !} \left[ \frac{d^{n}}{{d t}^{n}} {\left({t}^{2} - 1\right)}^{n} \right]_{t = z}

Integrals

e36542
11Pn ⁣(x)Pm ⁣(x)dx=22n+1δ(n,m)\int_{-1}^{1} P_{n}\!\left(x\right) P_{m}\!\left(x\right) \, dx = \frac{2}{2 n + 1} \delta_{(n,m)}

Sum representations

c5dd9b
Pn ⁣(z)=12nk=0n(nk)2(z1)nk(z+1)kP_{n}\!\left(z\right) = \frac{1}{{2}^{n}} \sum_{k=0}^{n} {{n \choose k}}^{2} {\left(z - 1\right)}^{n - k} {\left(z + 1\right)}^{k}
f0569a
Pn ⁣(z)=k=0n(nk)(n+kk)(z12)kP_{n}\!\left(z\right) = \sum_{k=0}^{n} {n \choose k} {n + k \choose k} {\left(\frac{z - 1}{2}\right)}^{k}
7a85b7
Pn ⁣(z)=12nk=0n/2(1)k(nk)(2n2kn)zn2kP_{n}\!\left(z\right) = \frac{1}{{2}^{n}} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose k} {2 n - 2 k \choose n} {z}^{n - 2 k}

Hypergeometric representations

9395fc
Pn ⁣(z)=2F1 ⁣(n,n+1,1,1z2)P_{n}\!\left(z\right) = \,{}_2F_1\!\left(-n, n + 1, 1, \frac{1 - z}{2}\right)
f55f0a
Pn ⁣(z)=(2nn)(z2)n2F1 ⁣(n2,1n2,12n,1z2)P_{n}\!\left(z\right) = {2 n \choose n} {\left(\frac{z}{2}\right)}^{n} \,{}_2F_1\!\left(-\frac{n}{2}, \frac{1 - n}{2}, \frac{1}{2} - n, \frac{1}{{z}^{2}}\right)
3c87b9
Pn ⁣(z)=(z12)n2F1 ⁣(n,n,1,z+1z1)P_{n}\!\left(z\right) = {\left(\frac{z - 1}{2}\right)}^{n} \,{}_2F_1\!\left(-n, -n, 1, \frac{z + 1}{z - 1}\right)
6cd4a1
P2n ⁣(z)=(1)n4n(2nn)2F1 ⁣(n,n+12,12,z2)P_{2 n}\!\left(z\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} {2 n \choose n} \,{}_2F_1\!\left(-n, n + \frac{1}{2}, \frac{1}{2}, {z}^{2}\right)
859445
P2n+1 ⁣(z)=(1)n4n(2n+1)(2nn)z2F1 ⁣(n,n+32,32,z2)P_{2 n + 1}\!\left(z\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} \left(2 n + 1\right) {2 n \choose n} z \,{}_2F_1\!\left(-n, n + \frac{3}{2}, \frac{3}{2}, {z}^{2}\right)

Bounds and inequalities

1ba9a5
Pn ⁣(x)1\left|P_{n}\!\left(x\right)\right| \le 1
155343
Pn ⁣(x)2I0 ⁣(2nx12)2e2nx1/2\left|P_{n}\!\left(x\right)\right| \le 2 I_{0}\!\left(2 n \sqrt{\frac{\left|x - 1\right|}{2}}\right) \le 2 {e}^{2 n \sqrt{\left|x - 1\right| / 2}}
ef4b53
Pn ⁣(z)Pn ⁣(zi)(z+1+z2)n\left|P_{n}\!\left(z\right)\right| \le \left|P_{n}\!\left(\left|z\right| i\right)\right| \le {\left(\left|z\right| + \sqrt{1 + {\left|z\right|}^{2}}\right)}^{n}
b786ad
Pn(x)n(n+1)2\left|P'_{n}(x)\right| \le \frac{n \left(n + 1\right)}{2}
60ac50
Pn(x)23/2πn1/2(1x2)3/4\left|P'_{n}(x)\right| \le \frac{{2}^{3 / 2}}{\sqrt{\pi}} \frac{{n}^{1 / 2}}{{\left(1 - {x}^{2}\right)}^{3 / 4}}
59e5df
Pn(x)(n1)n(n+1)(n+2)8\left|P''_{n}(x)\right| \le \frac{\left(n - 1\right) n \left(n + 1\right) \left(n + 2\right)}{8}
3b175b
Pn(x)25/2πn3/2(1x2)5/4\left|P''_{n}(x)\right| \le \frac{{2}^{5 / 2}}{\sqrt{\pi}} \frac{{n}^{3 / 2}}{{\left(1 - {x}^{2}\right)}^{5 / 4}}
6476bd
Pn(r)(x)2r+1/2πnr1/2(1x2)(2n+1)/4\left|{P}^{(r)}_{n}(x)\right| \le \frac{{2}^{r + 1 / 2}}{\sqrt{\pi}} \frac{{n}^{r - 1 / 2}}{{\left(1 - {x}^{2}\right)}^{\left( 2 n + 1 \right) / 4}}

Analytic properties

40fa59
Pn ⁣(z) is holomorphic on zCP_{n}\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C}
d36fd7
poleszC{~}Pn ⁣(z)={~}\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} P_{n}\!\left(z\right) = \left\{{\tilde \infty}\right\}
99e62f
EssentialSingularities ⁣(Pn ⁣(z),z,C{~})={}\operatorname{EssentialSingularities}\!\left(P_{n}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
7680d3
BranchPoints ⁣(Pn ⁣(z),z,C{~})={}\operatorname{BranchPoints}\!\left(P_{n}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
22a42f
BranchCuts ⁣(Pn ⁣(z),z,C)={}\operatorname{BranchCuts}\!\left(P_{n}\!\left(z\right), z, \mathbb{C}\right) = \left\{\right\}
415911
#zeroszCPn ⁣(z)=n\# \mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = n
df439e
zeroszCPn ⁣(z)(1,1)\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) \subset \left(-1, 1\right)
0745ee
zeroszCPn ⁣(z)={xn,1,,xn,n}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = \left\{x_{n,1}, \ldots, x_{n,n}\right\}
b2d723
Pn ⁣(z)=Pn ⁣(z)P_{n}\!\left(\overline{z}\right) = \overline{P_{n}\!\left(z\right)}

Gauss-Legendre quadrature

Related topics: Gaussian quadrature

ea4754
wn,k=2(1(xn,k)2)(Pn(xn,k))2w_{n,k} = \frac{2}{\left(1 - {\left(x_{n,k}\right)}^{2}\right) {\left(P'_{n}(x_{n,k})\right)}^{2}}
47b181
11f(t)dtk=1nwn,kf ⁣(xn,k)64M15(1ρ2)ρ2n   where M=suptEρf(t)\left|\int_{-1}^{1} f(t) \, dt - \sum_{k=1}^{n} w_{n,k} f\!\left(x_{n,k}\right)\right| \le \frac{64 M}{15 \left(1 - {\rho}^{-2}\right) {\rho}^{2 n}}\; \text{ where } M = \mathop{\operatorname{sup}}\limits_{t \in \mathcal{E}_{\rho}} \left|f(t)\right|

Bounds and inequalities

Turán's inequalities

c8d10e
(Pn ⁣(x))2Pn1 ⁣(x)Pn+1 ⁣(x)0{\left(P_{n}\!\left(x\right)\right)}^{2} - P_{n - 1}\!\left(x\right) P_{n + 1}\!\left(x\right) \ge 0
227d60
(Pn ⁣(x))2Pn1 ⁣(x)Pn+1 ⁣(x)>0{\left(P_{n}\!\left(x\right)\right)}^{2} - P_{n - 1}\!\left(x\right) P_{n + 1}\!\left(x\right) > 0

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-11-11 15:50:15.016492 UTC