Ferrers function

Ferrers function of the first kind P(x)

Ferrers function of the second kind Q(x)

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

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

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