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

$Q_v^{\mu }(x)=\frac{\pi }{2\sin (\mu \pi )}(\cos (\mu \pi ){\left(\frac{1+x}{1-x}\right)}^{\frac{\mu }{2}}\frac{{}_2{F}_1(v+1,-v;1-\mu ;\frac{1-x}{2})}{\Gamma (1-\mu )}-\frac{\Gamma (\nu +\mu +1)}{\Gamma (\nu -\mu +1)}{\left(\frac{1-x}{1+x}\right)}^{\frac{\mu }{2}}\frac{{}_2{F}_1(v+1,-v;1+\mu ;\frac{1-x}{2})}{\Gamma (1+\mu )})$

See also

• Legendre function