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, normalizations, 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	$P_n^{*}\left(x\right)$	$(0,1)$	$1$	$1/(2n+1)$	$2^{2n}{\left(\frac{1}{2}\right)}_n/n!$	$-\frac{1}{2}n$
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►For exact values of the coefficients of the Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$, the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$, the Chebyshev polynomials $T_n\left(x\right)$ and $U_n\left(x\right)$, the Legendre polynomials $P_n\left(x\right)$, the Laguerre polynomials $L_n\left(x\right)$, and the Hermite polynomials $H_n\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). … ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …

2: 18.30 Associated OP’s

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Associated Legendre Polynomials

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18.30.6 $$P_n(x;c)=P_n^{(0,0)}(x;c),$$ $n=0,1,\mathrm{\dots }$.

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Defines:

$P_n(x;c)$: associated Legendre polynomial

Symbols:

$P_n^{(\alpha ,\beta )}(x;c)$: associated Jacobi polynomial, $x$: real variable and $n$: continuous nonnegative real

Permalink:

http://dlmf.nist.gov/18.30.E6

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Annotations for §18.30, §18.30 and Ch.18

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18.30.7 $$P_n(x;c)=\sum _{\mathrm{\ell }=0}^n\frac{c}{\mathrm{\ell }+c}{P}_{\mathrm{\ell }}\left(x\right)P_{n-\mathrm{\ell }}\left(x\right).$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $P_n(x;c)$: associated Legendre polynomial, $\mathrm{\ell }$: nonnegative integer, $x$: real variable and $n$: continuous nonnegative real

Referenced by:

§18.30

Permalink:

http://dlmf.nist.gov/18.30.E7

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Annotations for §18.30, §18.30 and Ch.18

►(These polynomials are not to be confused with associated Legendre functions §14.3(ii).) ►For further results on associated Legendre polynomials see Chihara (1978, Chapter VI, §12); on associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991). …

3: 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). …

4: 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, $\mathrm{e}$: 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

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Annotations for §10.59 and Ch.10

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

5: 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

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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

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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

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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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6: 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

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

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14.7.11 $$P_n^m\left(x\right)={\left(x^2-1\right)}^{m/2}\frac{d^m}{{dx}^m}{P}_n\left(x\right),$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\frac{df}{dx}$: derivative, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.7.E11

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

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7: 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, $\mathrm{e}$: 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, $\mathrm{e}$: 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 $\nu =n$. …

8: 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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9: 18.13 Continued Fractions

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Legendre

► $P_n\left(x\right)$ is the denominator of the $n$th approximant to: …

10: 18.17 Integrals

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Legendre

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Legendre

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Legendre

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Legendre

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Legendre

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