Ferrers function

Ferrers Function First Kind P(x)

Ferrers Function Second Kind Q(x)

In mathematics, Ferrers functions are defined in terms of hypergeometric functions

Ferrers function of the first kind

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

Ferrers function of the second kind

$Q_{v}^{\mu }(x)=\cos(\mu \pi )\left({\frac {1+x}{1-x}}\right)^{\mu /2}{\frac {F(v+1,-v;1-\mu ;1/2-2/x)}{\Gamma (1-\mu )}}$

Relation with other special function

$P_{v}^{\mu }(x)=\left(-{\frac {1+x}{-1+x}}\right)^{\mu /2}\operatorname {HeunC} \left(0,-\mu ,2\,v+1,0,v+1/2+v^{2},{\frac {-1+x}{1+x}}\right)\left((1/2+1/2\,x)^{v+1}\right)^{-1}\left(\Gamma \left(1-\mu \right)\right)^{-1}$