Legendre symbol

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21: Bibliography S

… ►

J. Segura and A. Gil (1999) Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.

ⓘ

Notes:: Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)
External Links:: ISSN 1068-9613, FullText, MathReview (Adam Marlewski), ZentralBlatt
Referenced by:: §14.30(i), 3rd item.
Encodings:: BibTeX

…

22: 14.15 Uniform Asymptotic Approximations

… ►

14.15.1

\mathsf{P}^{-\mu}_{\nu}\left(\pm x\right)=\left(\frac{1\mp x}{1\pm x}\right)^{% \mu/2}\left(\sum_{j=0}^{J-1}\frac{{\left(\nu+1\right)_{j}}{\left(-\nu\right)_{% j}}}{j!\Gamma\left(j+1+\mu\right)}\left(\frac{1\mp x}{2}\right)^{j}+O\left(% \frac{1}{\Gamma\left(J+1+\mu\right)}\right)\right)

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $x$ : real variable, $\mu$ : general order, $\nu$ : general degree and $J$ : nonnegative integer
Referenced by:: §14.15(i)
Permalink:: http://dlmf.nist.gov/14.15.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.15(i), §14.15 and Ch.14

…

23: 18.11 Relations to Other Functions

… ►

18.11.1

\mathsf{P}^{m}_{n}\left(x\right)={\left(\tfrac{1}{2}\right)_{m}}(-2)^{m}(1-x^{% 2})^{\frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x\right)={\left(n+1\right)_{m% }}(-2)^{-m}(1-x^{2})^{\frac{1}{2}m}P^{(m,m)}_{n-m}\left(x\right),

0\leq m\leq n

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §14.3(iv), §18.11(i)
Permalink:: http://dlmf.nist.gov/18.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

…

F. G. Garvan and M. E. H. Ismail (Eds.) (2001) Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, Vol. 4, Kluwer Academic Publishers, Dordrecht.

ⓘ

External Links:: ISBN 1-4020-0101-0, MathReview Entry
Referenced by:: Profile Mourad E.H. Ismail.
Encodings:: BibTeX

… ►

W. Gautschi (1965) Algorithm 259: Legendre functions for arguments larger than one. Comm. ACM 8 (8), pp. 488–492.

ⓘ

Notes:: Variable-precision Algol. For remark see ACM Trans. Math. Software 3(2) (1977), p. 204–205
External Links:: ISSN 0001-0782, Document
Referenced by:: §14.34(ii).
Encodings:: BibTeX

… ►

A. Gervois and H. Navelet (1985a) Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.

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External Links:: ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

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A. Gervois and H. Navelet (1985b) Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.

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External Links:: ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

… ►

J. N. Ginocchio (1991) A new identity for some six-

j

symbols. J. Math. Phys. 32 (6), pp. 1430–1432.

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External Links:: ISSN 0022-2488, Document, MathReview
Referenced by:: §34.5(vi).
Encodings:: BibTeX

…

25: 10.54 Integral Representations

… ►

10.54.2

\mathsf{j}_{n}\left(z\right)=\frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos\theta}P% _{n}\left(\cos\theta\right)\sin\theta\,\mathrm{d}\theta.

ⓘ

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.14
Permalink:: http://dlmf.nist.gov/10.54.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.54 and Ch.10

►

10.54.3

\mathsf{k}_{n}\left(z\right)=\frac{\pi}{2}\int_{1}^{\infty}e^{-zt}P_{n}\left(t% \right)\,\mathrm{d}t,

|\operatorname{ph}z|<\tfrac{1}{2}\pi.

ⓘ

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.54.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §10.54 and Ch.10

►

10.54.4

\mathsf{j}_{n}\left(z\right)=\frac{(-i)^{n+1}}{2\pi}\int_{i\infty}^{(-1+,1+)}e% ^{izt}Q_{n}\left(t\right)\,\mathrm{d}t,

|\operatorname{ph}z|<\tfrac{1}{2}\pi.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $\operatorname{ph}$ : phase, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.54.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §10.54 and Ch.10

►

{\mathsf{h}^{(1)}_{n}}\left(z\right)=\frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(1+% )}e^{izt}Q_{n}\left(t\right)\,\mathrm{d}t,

… ►For the Legendre polynomial

P_{n}

and the associated Legendre function

Q_{n}

see §§18.3 and 14.21(i), with

\mu=0

and

\nu=n

. …

26: 10.60 Sums

… ►Then with

P_{n}

again denoting the Legendre polynomial of degree

n

, ►

10.60.1

\frac{\cos w}{w}=-\sum_{n=0}^{\infty}(2n+1)\mathsf{j}_{n}\left(v\right)\mathsf% {y}_{n}\left(u\right)P_{n}\left(\cos\alpha\right),

|ve^{\pm i\alpha}|<|u|

ⓘ

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind and $n$ : integer
A&S Ref:: 10.1.46 (corrected)
Referenced by:: §10.60(i)
Permalink:: http://dlmf.nist.gov/10.60.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §10.60(i), §10.60 and Ch.10

►

10.60.2

\frac{\sin w}{w}=\sum_{n=0}^{\infty}(2n+1)\mathsf{j}_{n}\left(v\right)\mathsf{% j}_{n}\left(u\right)P_{n}\left(\cos\alpha\right).

ⓘ

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind and $n$ : integer
A&S Ref:: 10.1.45
Referenced by:: §10.60(iii)
Permalink:: http://dlmf.nist.gov/10.60.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.60(i), §10.60 and Ch.10

… ►

10.60.7

e^{iz\cos\alpha}=\sum_{n=0}^{\infty}(2n+1)i^{n}\mathsf{j}_{n}\left(z\right)P_{% n}\left(\cos\alpha\right),

ⓘ

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.47
Referenced by:: §10.60(iii), Ch.10
Permalink:: http://dlmf.nist.gov/10.60.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §10.60(iii), §10.60 and Ch.10

►

10.60.8

e^{z\cos\alpha}=\sum_{n=0}^{\infty}(2n+1){\mathsf{i}^{(1)}_{n}}\left(z\right)P% _{n}\left(\cos\alpha\right),

ⓘ

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.2.36
Permalink:: http://dlmf.nist.gov/10.60.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §10.60(iii), §10.60 and Ch.10

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27: Bibliography R

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J. Raynal (1979) On the definition and properties of generalized

6

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

… ►

REDUCE (free interactive system)

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Notes:: Symbolic and extended-precision general purpose computer algebra system.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

L. Robin (1957) Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome I. Gauthier-Villars, Paris.

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Notes:: Préface de H. Villat
External Links:: MathReview (R. Campbell), ZentralBlatt
Referenced by:: Ch.14.
Encodings:: BibTeX

►

L. Robin (1958) Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome II. Gauthier-Villars, Paris.

ⓘ

External Links:: MathReview (R. Campbell), ZentralBlatt
Referenced by:: Ch.14.
Encodings:: BibTeX

… ►

C. C. J. Roothaan and S. Lai (1997) Calculation of

3n

j

symbols by Labarthe’s method. International Journal of Quantum Chemistry 63 (1), pp. 57–64.

ⓘ

External Links:: Document
Referenced by:: §34.13.
Encodings:: BibTeX

…

28: 18.12 Generating Functions

… ►The

z

-radii of convergence will depend on

x

, and in first instance we will assume

x\in[-1,1]

for Jacobi, ultraspherical, Chebyshev and Legendre,

x\in[0,\infty)

for Laguerre, and

x\in\mathbb{R}

for Hermite. … ►

Legendre

►

18.12.11

\frac{1}{\sqrt{1-2xz+z^{2}}}=\sum_{n=0}^{\infty}P_{n}\left(x\right)z^{n},

|z|<1

ⓘ

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : 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

►

18.12.12

{\mathrm{e}}^{xz}J_{0}\left(z\sqrt{1-x^{2}}\right)=\sum_{n=0}^{\infty}\frac{P_% {n}\left(x\right)}{n!}z^{n}.

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $P_{\NVar{n}}\left(\NVar{x}\right)$ : 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

… ►

18.12.14

\Gamma\left(\alpha+1\right)(xz)^{-\frac{1}{2}\alpha}{\mathrm{e}}^{z}J_{\alpha}% \left(2\sqrt{xz}\right)=\sum_{n=0}^{\infty}\frac{L^{(\alpha)}_{n}\left(x\right% )}{{\left(\alpha+1\right)_{n}}}z^{n}.

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.9.16
Referenced by:: §18.12
Permalink:: http://dlmf.nist.gov/18.12.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §18.12, §18.12 and Ch.18

…

29: 18.30 Associated OP’s

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§18.30(ii) Associated Legendre Polynomials

… ►

18.30.6

P_{n}\left(x;c\right)=P^{(0,0)}_{n}\left(x;c\right),

n=0,1,\dots

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Defines:: $P_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$ : associated Legendre polynomial
Symbols:: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$ : 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

… ►

18.30.7

P_{n}\left(x;c\right)=\sum_{\ell=0}^{n}\frac{c}{\ell+c}P_{\ell}\left(x\right)P% _{n-\ell}\left(x\right),

ⓘ

Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $P_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$ : associated Legendre polynomial, $\ell$ : 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). …

30: Software Index

… ► ►►►►►►►

	Open Source						With Book				Commercial
…
14 Legendre and Related Functions
14.34(ii) $\mathsf{P}_{\nu}\left(x\right)$ , $\mathsf{Q}_{\nu}\left(x\right)$ , $P_{\nu}\left(x\right)$ , $Q_{\nu}\left(x\right)$ , $x,\nu\in\mathbb{R}$	✓	✓	✓	✓	a	✓	✓	✓	✓	✓		✓	✓	✓	✓
14.34(iii) $\mathsf{P}_{\nu}\left(z\right)$ , $\mathsf{Q}_{\nu}\left(z\right)$ , $P_{\nu}\left(z\right)$ , $Q_{\nu}\left(z\right)$ , $z,\nu\in\mathbb{C}$	✓				a	✓	✓					✓	✓	a
14.34(iv) $\mathsf{P}_{-\frac{1}{2}+i\tau}\left(x\right)$ , $\mathsf{Q}_{-\frac{1}{2}+i\tau}\left(x\right)$ , $\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ , $P_{-\frac{1}{2}+i\tau}\left(x\right)$ , $Q_{-\frac{1}{2}+i\tau}\left(x\right)$		✓					✓		✓
…
34 3j, 6j, 9j Symbols
…

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