Ferrers function explained

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 μ and the degree ν are real and x ∈ (-1,1)

Ferrers function of the first kind

P ^{\mu(x)} _{v} = \left( (1+ x)/(1- x) \right) ^{\mu/2} ⋅ ({)/(_{2F}) _{1(v+1,-v;1-\mu;1/2-x/2)}{\Gamma(1-\mu)}}

Ferrers function of the second kind

Q ^{\mu(x) =} _{v} (\pi)/(\left(\cos(\mu\pi)\left( 2 \sin(\mu\pi)) (1+ x)/(1- x) \right) ^{(\mu)/(2 ({)/(_{2F}))} _{1 \left(v +1 , - v; 1- \mu; (1- x)/(2 \right)}{\Gamma( 1- \mu)} - (\Gamma(\nu + \mu +1 ))/(\Gamma(\nu - \mu +1 )))} \left( (1- x)/(1+ x) \right) ^{(\mu)/(2 ({)/(_{2F}))} _{1 \left(v +1 , - v; 1+ \mu; (1- x)/(2 \right)}{\Gamma( 1+ \mu)}\right))}

See also

• Legendre function