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

Encodings:

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See also:

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

Encodings:

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See also:

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: 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:

pMML, png, TeX

See also:

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

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4: 18.41 Tables

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

5: 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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6: 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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7: 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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Annotations for §18.10(i), §18.10(i), §18.10 and Ch.18

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

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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 $x$ , $\mathrm{i}$: imaginary unit, $\int$: 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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8: 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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$P_n\left(x\right)$	$1-x^2$	$-2x$	$0$	$n(n+1)$
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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: 18.7 Interrelations and Limit Relations

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Legendre, Ultraspherical, and Jacobi

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18.7.9 $$P_n\left(x\right)=C_n^{(\frac{1}{2})}\left(x\right)=P_n^{(0,0)}\left(x\right).$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $P_n\left(x\right)$: Legendre polynomial, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.35, 22.5.36

Referenced by:

§10.60(iii), §18.18(ii), §18.5(iii)

Permalink:

http://dlmf.nist.gov/18.7.E9

Encodings:

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See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

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18.7.10 $$P_n^{*}\left(x\right)=P_n\left(2x-1\right).$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $P_n^{*}\left(x\right)$: shifted Legendre polynomial, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.7.E10

Encodings:

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See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

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