relation to associated Legendre functions

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1: 14.3 Definitions and Hypergeometric Representations

§14.3 Definitions and Hypergeometric Representations

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§14.3(ii) Interval

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§14.3(iii) Alternative Hypergeometric Representations

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14.3.14 $$w_2(\nu ,\mu ,x)=\frac{2^{\mu }\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)}x{\left(1-x^2\right)}^{-\mu /2}𝐅(\frac{1}{2}-\frac{1}{2}\nu -\frac{1}{2}\mu ,\frac{1}{2}\nu -\frac{1}{2}\mu +1;\frac{3}{2};x^2).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $\mu$: general order, $\nu$: general degree and $w_2$: function $w_1$,

Referenced by:

§10.59, §14.3(i), §14.5(i)

Permalink:

http://dlmf.nist.gov/14.3.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

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§14.3(iv) Relations to Other Functions

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2: 14.7 Integer Degree and Order

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§14.7(i) $\mu =0$

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3: 14.5 Special Values

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§14.5(v) $\mu =0$, $\nu =\pm \frac{1}{2}$

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4: 15.9 Relations to Other Functions

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§15.9(iv) Associated Legendre Functions; Ferrers Functions

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5: 30.6 Functions of Complex Argument

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Relations to Associated Legendre Functions

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6: 14.2 Differential Equations

… ►Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations ${𝖯}_{\nu }^0\left(x\right)={𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }^0\left(x\right)={𝖰}_{\nu }\left(x\right)$, $P_{\nu }^0\left(x\right)=P_{\nu }\left(x\right)$, $Q_{\nu }^0\left(x\right)=Q_{\nu }\left(x\right)$, ${𝑸}_{\nu }^0\left(x\right)={𝑸}_{\nu }\left(x\right)=Q_{\nu }\left(x\right)/\mathrm{\Gamma }\left(\nu +1\right)$. …

7: 14.21 Definitions and Basic Properties

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§14.21(i) Associated Legendre Equation

… ►Standard solutions: the associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $P_{\nu }^{-\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{-\nu -1}^{\mu }\left(z\right)$. … ►

§14.21(ii) Numerically Satisfactory Solutions

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§14.21(iii) Properties

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8: 14 Legendre and Related Functions

Chapter 14 Legendre and Related Functions

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9: 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. Representations of special functions in terms of the Meijer $G$-function are given in Erdélyi et al. (1953a, §5.6), Luke (1969a, §§6.4–6.5), and Mathai (1993, §3.10).

10: 18.3 Definitions

§18.3 Definitions

… ►For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►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)). …