Legendre symbol

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11: 18.6 Symmetry, Special Values, and Limits to Monomials

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

18.6.1 $$L_n^{(\alpha )}\left(0\right)=\frac{{\left(\alpha +1\right)}_n}{n!}.$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$) and $n$: nonnegative integer

A&S Ref:

22.4.7 (see column for $f_n(0)$)

Referenced by:

§18.17(i), §18.6(i), §18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.6(i), §18.6(i), §18.6 and Ch.18

►

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

► ►►►

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

►

…

12: 10.59 Integrals

… ►

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:

pMML, png, TeX

See also:

Annotations for §10.59 and Ch.10

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

13: 18.3 Definitions

§18.3 Definitions

… ►This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … ►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). … ►

Legendre

►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …

14: 18.17 Integrals

… ►

18.17.38 $${\int }_0^1{P}_{2n}\left(x\right)x^{z-1}dx=\frac{{(-1)}^n{\left(\frac{1}{2}-\frac{1}{2}z\right)}_n}{2{\left(\frac{1}{2}z\right)}_{n+1}},$$ $\mathrm{\Re }z>0$,

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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18.17.39 $${\int }_0^1{P}_{2n+1}\left(x\right)x^{z-1}dx=\frac{{(-1)}^n{\left(1-\frac{1}{2}z\right)}_n}{2{\left(\frac{1}{2}+\frac{1}{2}z\right)}_{n+1}},$$ $\mathrm{\Re }z>-1$.

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

…

15: 18.5 Explicit Representations

§18.5 Explicit Representations

… ►For corresponding formulas for Chebyshev, Legendre, and the Hermite ${\mathrm{𝐻𝑒}}_n$ polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … ►

Legendre

►

$P_0\left(x\right)=1,$

… ►

16: 14.12 Integral Representations

… ►

14.12.7 $$P_{\nu }^m\left(x\right)=\frac{{\left(\nu +1\right)}_m}{\pi }{\int }_0^{\pi }{\left(x+{\left(x^2-1\right)}^{1/2}\cos \varphi \right)}^{\nu }\cos \left(m\varphi \right)d\varphi ,$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.12.E7

Encodings:

pMML, png, TeX

See also:

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

…

17: 14.17 Integrals

… ►

14.17.6 $${\int }_{-1}^1{𝖯}_l^m\left(x\right){𝖯}_n^m\left(x\right)dx=\frac{(n+m)!}{(n-m)!\left(n+\frac{1}{2}\right)}{\delta }_{l,n},$$

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Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

A&S Ref:

8.14.11 8.14.13

Referenced by:

§14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

Permalink:

http://dlmf.nist.gov/14.17.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

►

14.17.7 $${\int }_{-1}^1{𝖯}_l^m\left(x\right){𝖯}_n^{-m}\left(x\right)dx=\frac{{(-1)}^m}{l+\frac{1}{2}}{\delta }_{l,n},$$

ⓘ

Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

Referenced by:

§14.17(iii)

Permalink:

http://dlmf.nist.gov/14.17.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right and ${(-1)}^m$ has been moved to the numerator of the fraction.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

►

14.17.8 $${\int }_{-1}^1\frac{{𝖯}_n^l\left(x\right){𝖯}_n^m\left(x\right)}{1-x^2}dx=\frac{(n+m)!}{(n-m)!m}{\delta }_{l,m},$$ $m>0$,

ⓘ

Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

A&S Ref:

8.14.12 8.14.14

Referenced by:

§14.17(iii)

Permalink:

http://dlmf.nist.gov/14.17.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

►

14.17.9 $${\int }_{-1}^1\frac{{𝖯}_n^l\left(x\right){𝖯}_n^{-m}\left(x\right)}{1-x^2}dx=\frac{{(-1)}^l}{l}{\delta }_{l,m},$$ $l>0$.

ⓘ

Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

Referenced by:

§14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

Permalink:

http://dlmf.nist.gov/14.17.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right and ${(-1)}^l$ has been moved to the numerator of the fraction.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

…

18: 14.13 Trigonometric Expansions

… ►

14.13.1 $${𝖯}_{\nu }^{\mu }\left(\cos \theta \right)=\frac{2^{\mu +1}{(\sin \theta )}^{\mu }}{{\pi }^{1/2}}\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\nu +\mu +k+1\right)}{\mathrm{\Gamma }\left(\nu +k+\frac{3}{2}\right)}\frac{{\left(\mu +\frac{1}{2}\right)}_k}{k!}\sin \left((\nu +\mu +2k+1)\theta \right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : variable

Source:

Volkmer (2021)

A&S Ref:

8.7.1

Referenced by:

§14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2)

Permalink:

http://dlmf.nist.gov/14.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

►

14.13.2 $${𝖰}_{\nu }^{\mu }\left(\cos \theta \right)={\pi }^{1/2}{2}^{\mu }{(\sin \theta )}^{\mu }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\nu +\mu +k+1\right)}{\mathrm{\Gamma }\left(\nu +k+\frac{3}{2}\right)}\frac{{\left(\mu +\frac{1}{2}\right)}_k}{k!}\cos \left((\nu +\mu +2k+1)\theta \right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : variable

Source:

Cohl et al. (2021, Theorem 6.2)

A&S Ref:

8.7.2

Referenced by:

§14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2)

Permalink:

http://dlmf.nist.gov/14.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

…

19: 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). …

20: 14.8 Behavior at Singularities

… ►

14.8.2 $${𝖯}_{\nu }^m\left(x\right)\sim {(-1)}^m\frac{{\left(\nu -m+1\right)}_{2m}}{m!}{\left(\frac{1-x}{2}\right)}^{m/2},$$ $m=1,2,3,\mathrm{\dots }$, $\nu \ne m-1,m-2,\mathrm{\dots },-m$,

ⓘ

Symbols:

: Ferrers function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.30(ii), §14.8(i)

Permalink:

http://dlmf.nist.gov/14.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(i), §14.8 and Ch.14

►

14.8.3 $${𝖰}_{\nu }\left(x\right)=\frac{1}{2}\ln \left(\frac{2}{1-x}\right)-\gamma -\psi \left(\nu +1\right)+O\left(\left(1-x\right)\ln \left(1-x\right)\right),$$ $\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable and $\nu$: general degree

Referenced by:

§14.8(i), Erratum (V1.1.5) for Equation (14.8.3)

Permalink:

http://dlmf.nist.gov/14.8.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.1.5):

The symbol $O\left(1-x\right)$ has been corrected to be $O\left(\left(1-x\right)\ln \left(1-x\right)\right)$.

Suggested 2022-02-08 by Mark Ashbaugh

See also:

Annotations for §14.8(i), §14.8 and Ch.14

… ►

14.8.9 $${𝑸}_{\nu }\left(x\right)=-\frac{\ln \left(x-1\right)}{2\mathrm{\Gamma }\left(\nu +1\right)}+\frac{\frac{1}{2}\ln 2-\gamma -\psi \left(\nu +1\right)}{\mathrm{\Gamma }\left(\nu +1\right)}+O\left(\left(x-1\right)\ln \left(x-1\right)\right),$$ $\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $\nu$: general degree

Referenced by:

Erratum (V1.1.5) for Equation (14.8.9)

Permalink:

http://dlmf.nist.gov/14.8.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.1.5):

The symbol $O\left(x-1\right)$ has been corrected to be $O\left(\left(x-1\right)\ln \left(x-1\right)\right)$.

Suggested 2022-02-09 by Mark Ashbaugh

See also:

Annotations for §14.8(ii), §14.8 and Ch.14

…