Legendre polynomials

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1: 18.3 Definitions

§18.3 Definitions

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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
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Legendre	$P_n\left(x\right)$	$(-1,1)$	$1$	$2/(2n+1)$	$2^n{\left(\frac{1}{2}\right)}_n/n!$	$0$
Shifted Legendre	$P_n^{\ast }\left(x\right)$	$(0,1)$	$1$	$1/(2n+1)$	$2^{2n}{\left(\frac{1}{2}\right)}_n/n!$	$-\frac{1}{2}n$
…

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►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). … ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …

2: 18.41 Tables

… ►For $P_n\left(x\right)$ ($={𝖯}_n\left(x\right)$) see §14.33. … ►For $P_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ see §3.5(v). …

3: 10.59 Integrals

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10.59.1

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind and $n$: integer

A&S Ref:

11.4.26

Referenced by:

§10.59

Permalink:

http://dlmf.nist.gov/10.59.E1

Encodings:

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See also:

Annotations for §10.59 and Ch.10

►where $P_n$ is the Legendre polynomial (§18.3). …

4: 10.60 Sums

… ►Then with $P_n$ again denoting the Legendre polynomial of degree $n$, ►

10.60.1 $$\frac{\cos w}{w}=-\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗃}_n\left(v\right){𝗒}_n\left(u\right)P_n\left(\cos \alpha \right),$$ .

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind and $n$: integer

A&S Ref:

10.1.46 (corrected)

Referenced by:

§10.60(i)

Permalink:

http://dlmf.nist.gov/10.60.E1

Encodings:

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Annotations for §10.60(i), §10.60 and Ch.10

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10.60.2 $$\frac{\sin w}{w}=\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗃}_n\left(v\right){𝗃}_n\left(u\right)P_n\left(\cos \alpha \right).$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind and $n$: integer

A&S Ref:

10.1.45

Referenced by:

§10.60(iii)

Permalink:

http://dlmf.nist.gov/10.60.E2

Encodings:

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See also:

Annotations for §10.60(i), §10.60 and Ch.10

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10.60.7 $${\mathrm{e}}^{\mathrm{i}z\cos \alpha }=\sum _{n=0}^{\mathrm{\infty }}(2n+1){\mathrm{i}}^n{𝗃}_n\left(z\right)P_n\left(\cos \alpha \right),$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.47

Referenced by:

§10.60(iii), Ch.10

Permalink:

http://dlmf.nist.gov/10.60.E7

Encodings:

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See also:

Annotations for §10.60(iii), §10.60 and Ch.10

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10.60.8 $${\mathrm{e}}^{z\cos \alpha }=\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗂}_n^{(1)}\left(z\right)P_n\left(\cos \alpha \right),$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

A&S Ref:

10.2.36

Permalink:

http://dlmf.nist.gov/10.60.E8

Encodings:

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See also:

Annotations for §10.60(iii), §10.60 and Ch.10

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5: 14.7 Integer Degree and Order

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§14.7(i) $\mu =0$

… ►where $P_n\left(x\right)$ is the Legendre polynomial of degree $n$. For additional properties of $P_n\left(x\right)$ see Chapter 18. … ►

14.7.4 $$W_{n-1}(x)=\sum _{k=1}^n\frac{1}{k}{P}_{k-1}\left(x\right)P_{n-k}\left(x\right).$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $x$: real variable, $n$: nonnegative integer and $W_{n-1}(x)$: quantity

A&S Ref:

8.6.19

Permalink:

http://dlmf.nist.gov/14.7.E4

Encodings:

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See also:

Annotations for §14.7(i), §14.7 and Ch.14

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14.7.13 $$P_n\left(x\right)=\frac{1}{2^{n}n!}\frac{d^n}{{dx}^n}{\left(x^2-1\right)}^n,$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $!$: factorial (as in $n!$), $x$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/14.7.E13

Encodings:

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See also:

Annotations for §14.7(ii), §14.7 and Ch.14

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6: 10.54 Integral Representations

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10.54.2 $${𝗃}_n\left(z\right)=\frac{{(-\mathrm{i})}^n}{2}{\int }_0^{\pi }{\mathrm{e}}^{\mathrm{i}z\cos \theta }{P}_n\left(\cos \theta \right)\sin \theta d\theta .$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.14

Permalink:

http://dlmf.nist.gov/10.54.E2

Encodings:

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See also:

Annotations for §10.54 and Ch.10

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10.54.3 $${𝗄}_n\left(z\right)=\frac{\pi }{2}{\int }_1^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{P}_n\left(t\right)dt,$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.54.E3

Encodings:

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See also:

Annotations for §10.54 and Ch.10

… ►For the Legendre polynomial $P_n$ and the associated Legendre function $Q_n$ see §§18.3 and 14.21(i), with $\mu =0$ and $\nu =n$. …

7: 18.4 Graphics

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Figure 18.4.4: Legendre polynomials $P_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

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8: 18.6 Symmetry, Special Values, and Limits to Monomials

… ►For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1. … ►

Table 18.6.1: Classical OP’s: symmetry and special values.

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$p_n(x)$	$p_n(-x)$	$p_n(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime }(0)$
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$P_n\left(x\right)$	${(-1)}^n{P}_n\left(x\right)$	$1$	${(-1)}^n{\left(\frac{1}{2}\right)}_n/n!$	$2{(-1)}^n{\left(\frac{1}{2}\right)}_{n+1}/n!$
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9: 18.10 Integral Representations

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Legendre

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18.10.2 $$P_n\left(\cos \theta \right)=\frac{2^{\frac{1}{2}}}{\pi }{\int }_0^{\theta }\frac{\cos \left((n+\frac{1}{2})\varphi \right)}{{(\cos \varphi -\cos \theta )}^{\frac{1}{2}}}d\varphi ,$$ .

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

A&S Ref:

22.10.13

Referenced by:

§18.10(i)

Permalink:

http://dlmf.nist.gov/18.10.E2

Encodings:

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See also:

Annotations for §18.10(i), §18.10(i), §18.10 and Ch.18

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18.10.5 $$P_n\left(\cos \theta \right)=\frac{1}{\pi }{\int }_0^{\pi }{(\cos \theta +i\sin \theta \cos \varphi )}^{n}d\varphi .$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $n$: nonnegative integer

A&S Ref:

22.10.12

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E5

Encodings:

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See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

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Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).

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$p_n(x)$	$g_0(x)$	$g_1(z,x)$	$g_2(z,x)$	$c$	Conditions
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$P_n\left(x\right)$	$1$	$z^{-1}$	${(1-2xz+z^2)}^{-\frac{1}{2}}$	$0$
$P_n\left(x\right)$	$1$	${\displaystyle \frac{z^2-1}{2(z-x)}}$	$1$	$x$
…

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10: 18.8 Differential Equations

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Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

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#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
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7	$P_n\left(x\right)$	$1-x^2$	$-2x$	$0$	$n(n+1)$
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