Ferrers function

In mathematics, Ferrers functions are certain special functions defined in terms of hypergeometric functions. They are named after Norman Macleod Ferrers.

Definitions

Define $\mu$ the order, and the $\nu$ degree are real, and assume $x\in (-1,+1)$.

Ferrers function of the first kind

$P_{v}^{\mu }(x)=\left({\frac {1+x}{1-x}}\right)^{\mu /2}\cdot {\frac {{}_{2}F_{1}(v+1,-v;1-\mu ;1/2-x/2)}{\Gamma (1-\mu )}}$

Ferrers function of the second kind

${\displaystyle Q_{v}^{\mu }(x)={\frac {\pi }{2\sin(\mu \pi )}}\left(\cos(\mu \pi )\left({\frac {1+x}{1-x}}\right)^{\frac {\mu }{2}}\,{\frac {{}_{2}F_{1}\left(v+1,-v;1-\mu$ ;{\frac {1-x}{2}}\right)}{\Gamma (1-\mu )}}-{\frac {\Gamma (\nu +\mu +1)}{\Gamma (\nu -\mu +1)}}\left({\frac {1-x}{1+x}}\right)^{\frac {\mu }{2}}\,{\frac {{}_{2}F_{1}\left(v+1,-v;1+\mu ;{\frac {1-x}{2}}\right)}{\Gamma (1+\mu )}}\right)}

See also

• Legendre function