quadratic Legendre symbol

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1: 27.9 Quadratic Characters

§27.9 Quadratic Characters

►For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. …If $p$ does not divide $n$, then $(n|p)$ has the value $1$ when the quadratic congruence $x^2\equiv n(modp)$ has a solution, and the value $-1$ when this congruence has no solution. … ►

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

… ►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. …

2: 18.7 Interrelations and Limit Relations

… ►

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

►

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

… ►

§18.7(ii) Quadratic Transformations

… ►Equations (18.7.13)–(18.7.20) are special cases of (18.2.22)–(18.2.23). …

3: Bibliography D

… ►

Texas Instruments, Inc..

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Notes:

Symbolic and extended-precision numerical computing system.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

… ►

Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters. J. Number Theory 25 (1), pp. 72–80.

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External Links:

ISSN 0022-314X, Document, MathReview (T. Metsänkylä)

Referenced by:

§24.12(iii).

Encodings:

BibTeX

… ►

Novel identities for simple $n$-$j$ symbols. J. Mathematical Phys. 16, pp. 318–319.

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External Links:

Document, MathReview (M. J. Westwater)

Referenced by:

§34.5(vi).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.

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External Links:

ISSN 0308-2105, Document, MathReview, ZentralBlatt

Referenced by:

§14.15(i), §14.15(i), §14.15(ii), §14.15(ii), §14.26.

Encodings:

BibTeX

… ►

The Legendre Relation for Elliptic Integrals. In Paul Halmos: Celebrating 50 Years of Mathematics, J. H. Ewing and F. W. Gehring (Eds.), pp. 305–315.

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External Links:

ISBN 0-387-97509-8, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

BibTeX

…

4: 15.9 Relations to Other Functions

… ►

Legendre

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15.9.7 $$P_n\left(x\right)=F(\genfrac{}{}{0pt}{}{-n,n+1}{1};\frac{1-x}{2}).$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $P_n\left(x\right)$: Legendre polynomial, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

… ►

§15.9(iv) Associated Legendre Functions; Ferrers Functions

►Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. … ►The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers. …