Table of Integrals, Series, and Products, 8th Edition

8.75 Special cases and particular values

8.751

1. 

$P_{\nu }^m(x)={(-1)}^m\frac{\Gamma (\nu +m+1){\left(1-x^2\right)}^{\frac{m}{2}}}{2^m\Gamma (\nu -m+1)m!}F\left(m-\nu ,m+\nu +1;m+1;\frac{1-x}{2}\right)$

  MO 84

2. 

$P_{\nu }^m(z)=\frac{\Gamma (\nu +m+1){\left(z^2-1\right)}^{\frac{m}{2}}}{2^{m}m!\Gamma (\nu -m+1)}F\left(m-\nu ,m+\nu +1;m+1;\frac{1-z}{2}\right)$

  MO 84

3.⁸ 

$Q_{n+\frac{1}{2}}^{\mu }(z)=\frac{e^{\mu \text{π}i}\Gamma \left(\mu +n+\frac{3}{2}\right)}{2^{n+\frac{3}{2}}(n+1)!}{(z^2-1)}^{\frac{\mu }{2}}{\text{π}}^{1/2}{z}^{-n-\mu -3/2}F\left(\frac{\mu +n+\frac{5}{2}}{2},\frac{\mu +n+\frac{3}{2}}{2};n+2;\frac{1}{z^2}\right)$

  MO 84

8.752

1. 

$P_{\nu }^m(x)={(-1)}^m{\left(1-x^2\right)}^{\frac{m}{2}}\frac{d^m}{d{x}^m}{P}_{\nu }(x)$

  WH, MO 84, EH I ...