# relation to associated Legendre functions

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##### 1: 14.3 Definitions and Hypergeometric Representations
###### §14.3(iii) Alternative Hypergeometric Representations
14.3.14 $w_{2}(\nu,\mu,x)=\frac{2^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1% \right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)}x\left(1% -x^{2}\right)^{-\mu/2}\mathbf{F}\left(\tfrac{1}{2}-\tfrac{1}{2}\nu-\tfrac{1}{2% }\mu,\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1;\tfrac{3}{2};x^{2}\right).$
##### 6: 14.2 Differential Equations
Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations $\mathsf{P}^{0}_{\nu}\left(x\right)=\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}^{0}_{\nu}\left(x\right)=\mathsf{Q}_{\nu}\left(x\right)$, $P^{0}_{\nu}\left(x\right)=P_{\nu}\left(x\right)$, $Q^{0}_{\nu}\left(x\right)=Q_{\nu}\left(x\right)$, $\boldsymbol{Q}^{0}_{\nu}\left(x\right)=\boldsymbol{Q}_{\nu}\left(x\right)=Q_{% \nu}\left(x\right)/\Gamma\left(\nu+1\right)$. …
##### 7: 14.21 Definitions and Basic Properties
###### §14.21(i) AssociatedLegendre Equation
Standard solutions: the associated Legendre functions $P^{\mu}_{\nu}\left(z\right)$, $P^{-\mu}_{\nu}\left(z\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$, and $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(z\right)$. …
##### 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 functions14.7(i)). …