Ferrers function

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

Definitions
When the order &mu; and the degree &nu; are real and x ∈ (-1,1)
 * Ferrers function of the first kind


 * $$P_v^\mu(x) = \left(\frac{1+x}{1-x}\right)^{\mu/2}\cdot\frac{{}_2F_1(v+1,-v;1-\mu;1/2-x/2)}{\Gamma(1-\mu)}                                       $$


 * Ferrers function of the second kind


 * $$Q_v^\mu(x)= \frac{\pi}{2\sin(\mu\pi)}\left(\cos(\mu\pi)\left(\frac{1+x}{1-x}\right)^\frac{\mu}2\,\frac{{}_2F_1\left(v+1,-v;1-\mu;\frac{1-x}2\right)}{\Gamma(1-\mu)}-\frac{\Gamma(\nu+\mu+1)}{\Gamma(\nu-\mu+1)}\left(\frac{1-x}{1+x}\right)^\frac{\mu}2\,\frac{{}_2F_1\left(v+1,-v;1+\mu;\frac{1-x}2\right)}{\Gamma(1+\mu)}\right)$$