1. Differential Equation[-]

The function $${Q_{v}^{u}\left(x\right)}$$ satisfies the differential equation$$\displaystyle \left( -{x}^{2}+1 \right) ^{2}{\frac {d^{2}}{d{x}^{2}}}y \left( x \right) -2\,x \left( -{x}^{2}+1 \right) {\frac {d}{dx}}y \left( x \right) + \left( v \left( v+1 \right) \left( -{x}^{2}+1 \right) -{u}^{2} \right) y \left( x \right) = 0$$ with initial values $$y \left( 0 \right) =\frac{{2}^{u-1}\sqrt {\pi }\,\Gamma \left( 1/2+1/2\,u+1/2\,v \right) }{{{\rm e}^{1/2\,i\pi \, \left( -2\,u+v+1 \right) }}\,\Gamma \left( 1+1/2\,v-1/2\,u \right) }$$ and $$y' \left( 0 \right) =\frac{\sqrt {\pi }{2}^{u}\,\Gamma \left( 1+1/2\,u+1/2\,v \right) }{{{\rm e}^{1/2\,i\pi \, \left( -2\,u+v \right) }}\,\Gamma \left( 1/2\,v-1/2\,u+1/2 \right) }$$.
All formulas on this page are valid under the condition that $$u$$ and $$2\,v$$ are not integers (special values for parameters can be entered at the bottom).

4. Local Expansions at Singularities and at Infinity[-]

The differential equation above has 2 non-zero finite .

6. Parameters[-]

The associated Legendre function of the second kind $${Q_{v}^{u}\left(x\right)}$$ depends on the parameters $$v$$ and $$u$$. The boxes below can be used to rename or instantiate these parameters.
p1 =  p2 =