Ferrers function

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

Definitions

Ferrers function of the first kind

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

Ferrers function of the second kind

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

See also

• Legendre function