Ferrers function

Ferrers Function First Kind P(x)

Ferrers Function Second Kind Q(x)

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

Failed to parse (syntax error): {\displaystyle P_v^\mu(x)=\left( -{\frac {1+x}{-1+x}} \right) ^{1/2\,\mu}{\\operatorname{HeunC}} \left( 0,- \mu,2\,v+1,0,v+1/2+{v}^{2},{\frac {-1+x}{1+x}} \right) \left( \left( 1/2+1/2\,x \right) ^{v+1} \right) ^{-1} \left( \Gamma \left( 1-\mu \right) \right) ^{-1} }