Legendre polynomials

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11: 34.3 Basic Properties: $\mathit{3}j$ Symbol

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§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

►For the polynomials $P_l$ see §18.3, and for the function $Y_{l,m}$ see §14.30. ►

34.3.19

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, : $\mathit{3}j$ symbol and $l,l_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§14.18(iv), §14.30(iii), §34.3(vii)

Permalink:

http://dlmf.nist.gov/34.3.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §34.3(vii), §34.3 and Ch.34

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34.3.21

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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, : $\mathit{3}j$ symbol and $l,l_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§14.17(iv)

Permalink:

http://dlmf.nist.gov/34.3.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §34.3(vii), §34.3 and Ch.34

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12: 18.13 Continued Fractions

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Legendre

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

13: 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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14: 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:

pMML, png, TeX

See also:

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

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

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Symbols:

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

Permalink:

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

Encodings:

pMML, png, TeX

See also:

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

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15: 18.5 Explicit Representations

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Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).

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$p_n(x)$	$w(x)$	$F(x)$	${\kappa }_n$
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$P_n\left(x\right)$	$1$	$1-x^2$	${(-2)}^{n}n!$
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$P_0\left(x\right)=1,$

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$P_1\left(x\right)=x,$

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$P_2\left(x\right)=\frac{3}{2}{x}^2-\frac{1}{2},$

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$P_3\left(x\right)=\frac{5}{2}{x}^3-\frac{3}{2}x,$

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16: 18.1 Notation

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Classical OP’s

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Legendre: $P_n\left(x\right)$.

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Shifted Legendre: $P_n^{\ast }\left(x\right)$.

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17: 18.30 Associated OP’s

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§18.30(ii) 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, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.30

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(ii), §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, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.30

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(ii), §18.30 and Ch.18

►in which $P_n\left(x\right)$ are the Legendre polynomials of Table 18.3.1. ►For further results on associated Legendre polynomials see Chihara (1978, Chapter VI, §12). …

18: 14.12 Integral Representations

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14.12.5 $$P_{\nu }^{-\mu }\left(x\right)=\frac{{\left(x^2-1\right)}^{-\mu /2}}{\mathrm{\Gamma }\left(\mu \right)}{\int }_1^x{P}_{\nu }\left(t\right){(x-t)}^{\mu -1}dt,$$ $\mathrm{\Re }\mu >0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.17(i)

Permalink:

http://dlmf.nist.gov/14.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12 and Ch.14

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14.12.13 $${𝑸}_n\left(x\right)=\frac{1}{2(n!)}{\int }_{-1}^1\frac{P_n\left(t\right)}{x-t}dt.$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $n$: nonnegative integer

A&S Ref:

8.8.3 (modified)

Permalink:

http://dlmf.nist.gov/14.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12(ii), §14.12 and Ch.14

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19: 18.9 Recurrence Relations and Derivatives

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Table 18.9.1: Classical OP’s: recurrence relations (18.9.1).

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$p_n(x)$	$A_n$	$B_n$	$C_n$
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$P_n\left(x\right)$	$\frac{2n+1}{n+1}$	$0$	$\frac{n}{n+1}$
$P_n^{\ast }\left(x\right)$	$\frac{4n+2}{n+1}$	$-\frac{2n+1}{n+1}$	$\frac{n}{n+1}$
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Table 18.9.2: Classical OP’s: recurrence relations (18.9.2_1).

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$p_n(x)$	$a_n$	$b_n$	$c_n$
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$P_n\left(x\right)$	$\frac{n+1}{2n+1}$	$0$	$\frac{n}{2n+1}$
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20: 18.12 Generating Functions

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Legendre

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18.12.11 $$\frac{1}{\sqrt{1-2xz+z^2}}=\sum _{n=0}^{\mathrm{\infty }}{P}_n\left(x\right)z^n,$$ .

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.12

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.12 $${\mathrm{e}}^{xz}{J}_0\left(z\sqrt{1-x^2}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n\left(x\right)}{n!}{z}^n.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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