Legendre symbol

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1: 27.9 Quadratic Characters

… ►For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. If $p$ divides $n$, then the value of $(n|p)$ is $0$. …The Legendre symbol $(n|p)$, as a function of $n$, is a Dirichlet character (mod $p$). … ►

27.9.2 $$(2|p)={(-1)}^{(p^2-1)/8}.$$

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Symbols:

$(n|p)$: Legendre symbol and $p$: odd prime

Referenced by:

§27.9

Permalink:

http://dlmf.nist.gov/27.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.9 and Ch.27

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27.9.3 $$(p|q)(q|p)={(-1)}^{(p-1)(q-1)/4}.$$

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Symbols:

$(n|p)$: Legendre symbol, $p$: odd prime and $q$: odd prime

Referenced by:

§27.9

Permalink:

http://dlmf.nist.gov/27.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.9 and Ch.27

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2: 27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
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$(n|p)$	Legendre symbol; see §27.9.

3: 19.15 Advantages of Symmetry

… ►These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). …

4: 34.3 Basic Properties: $\mathit{3}j$ Symbol

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

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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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5: 14.6 Integer Order

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14.6.5 $${\left(\nu +1\right)}_m{𝑸}_{\nu }^m\left(x\right)={(-1)}^m{\left(x^2-1\right)}^{m/2}\frac{d^m{𝑸}_{\nu }\left(x\right)}{{dx}^m}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.21(iii), §14.6(i)

Permalink:

http://dlmf.nist.gov/14.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.6(i), §14.6 and Ch.14

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6: 19.12 Asymptotic Approximations

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19.12.1 $$K\left(k\right)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\left(\frac{1}{2}\right)}_m}{m!m!}k_{}^{\prime }{}_{}{}^{2m}\left(\ln \left(\frac{1}{k^{\prime }}\right)+d(m)\right),$$ ,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $d(m)$: function

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

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19.12.2 $$E\left(k\right)=1+\frac{1}{2}\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\left(\frac{3}{2}\right)}_m}{{\left(2\right)}_{m}m!}k_{}^{\prime }{}_{}{}^{2m+2}\left(\ln \left(\frac{1}{k^{\prime }}\right)+d(m)-\frac{1}{(2m+1)(2m+2)}\right),$$ ,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $d(m)$: function

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

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7: Errata

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The Legendre polynomial $P_n$ was mistakenly identified as the associated Legendre function $P_n$ in §§10.54, 10.59, 10.60, 18.18, 18.41, 34.3 (and was thus also affected by the bug reported below). These symbols now link correctly to their definitions. Reported by Roy Hughes on 2022-05-23

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

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

Reported by Mark Ashbaugh on 2022-02-08

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

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

Reported by Mark Ashbaugh on 2022-02-08

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8: 10 Bessel Functions

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9: 18 Orthogonal Polynomials

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10: 19.5 Maclaurin and Related Expansions

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19.5.1 $$K\left(k\right)=\frac{\pi }{2}\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\left(\frac{1}{2}\right)}_m}{m!m!}{k}^{2m}=\frac{\pi }{2}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2},\frac{1}{2}}{1};k^2),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $!$: factorial (as in $n!$), $m$: nonnegative integer and $k$: real or complex modulus

Source:

Carlson (1961b, (2.4))

Referenced by:

§19.36(iv), §19.5

Permalink:

http://dlmf.nist.gov/19.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.2 $$E\left(k\right)=\frac{\pi }{2}\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{2}\right)}_m{\left(\frac{1}{2}\right)}_m}{m!m!}{k}^{2m}=\frac{\pi }{2}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2},\frac{1}{2}}{1};k^2),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $!$: factorial (as in $n!$), $m$: nonnegative integer and $k$: real or complex modulus

Source:

Carlson (1961b, (2.5))

Referenced by:

§19.36(iv)

Permalink:

http://dlmf.nist.gov/19.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.4_1 $$F(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.8))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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19.5.4_2 $$E(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},-\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.9))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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19.5.4_3 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{F}_1(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};{\sin }^2\varphi ,{\alpha }^2{\sin }^2\varphi )k^{2m},$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Source:

Carlson (1961b, (2.11))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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