Legendre relation

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31: 14.17 Integrals

… ►Orthogonality relations for the associated Legendre functions of imaginary order are given in Bielski (2013). …

32: Bibliography B

… ►

Orthogonality relations for the associated Legendre functions of imaginary order. Integral Transforms Spec. Funct. 24 (4), pp. 331–337.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Praveen Agarwal), ZentralBlatt

Referenced by:

§14.17(iii), §14.17(iii).

Encodings:

BibTeX

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33: 22.8 Addition Theorems

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§22.8(iii) Special Relations Between Arguments

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22.8.22 $$z_1+z_2+z_3+z_4=2K\left(k\right).$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(iii), §22.8 and Ch.22

… ►For these and related identities see Copson (1935, pp. 415–416). ►If sums/differences of the $z_j$’s are rational multiples of $K\left(k\right)$, then further relations follow. … ►

22.8.24 $$z_1-z_2=z_2-z_3=\frac{2}{3}K\left(k\right),$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(iii), §22.8 and Ch.22

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34: 18.11 Relations to Other Functions

§18.11 Relations to Other Functions

… ►See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions. ►

Ultraspherical

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Laguerre

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Hermite

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35: 18.3 Definitions

§18.3 Definitions

… ►This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … ►It is also related to a discrete Fourier-cosine transform, see Britanak et al. (2007). ►

Legendre

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

36: 19.36 Methods of Computation

… ►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). …

37: 14.8 Behavior at Singularities

… ►In the next three relations $\mathrm{\Re }\mu >0$. … ►The behavior of ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ as $x\to -1+$ follows from the above results and the connection formulas (14.9.8) and (14.9.10). … ►

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:

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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

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14.8.10 $${𝑸}_{-n}\left(x\right)\to {(-1)}^{n+1}(n-1)!,$$ $n=1,2,3,\mathrm{\dots }$,

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Symbols:

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

Referenced by:

§14.8(ii)

Permalink:

http://dlmf.nist.gov/14.8.E10

Encodings:

pMML, png, TeX

See also:

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

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38: 14.31 Other Applications

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§14.31(ii) Conical Functions

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§14.31(iii) Miscellaneous

►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). … ►Legendre functions $P_{\nu }\left(x\right)$ of complex degree $\nu$ appear in the application of complex angular momentum techniques to atomic and molecular scattering (Connor and Mackay (1979)).

39: 3.5 Quadrature

… ►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … ►

Gauss–Legendre Formula

… ►The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi). … ►are related to Bessel polynomials (§§10.49(ii) and 18.34). … …

40: 16.18 Special Cases

§16.18 Special Cases

… ►This is a consequence of the following relations: …As a corollary, special cases of the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer $G$-function. …