1. Differential Equation[-]

The function $${P_{n}^{(a,b)}\left(x\right)}$$ satisfies the differential equation$$\displaystyle \left( {x}^{2}-1 \right) {\frac {d^{2}}{d{x}^{2}}}y \left( x \right) + \left( a-b+ \left( a+b+2 \right) x \right) {\frac {d}{dx}}y \left( x \right) -n \left( n+a+b+1 \right) y \left( x \right) = 0$$ with initial values $$y \left( 0 \right) =\frac{\Gamma \left( 1+a+n \right) \,{{}_{2}F_{1}\!\left(-n,-b-n;\,a+1;\,-1\right)}}{{2}^{n}\,\Gamma \left( a+1 \right) \,\Gamma \left( n+1 \right) }$$ and $$y' \left( 0 \right) =\frac{n\left(n+a+b+1\right)\Gamma \left( 1+a+n \right) \,{{}_{2}F_{1}\!\left(-b-n,1-n;\,a+2;\,-1\right)}}{{2}^{n}\left(a+1\right)\Gamma \left( a+1 \right) \,\Gamma \left( n+1 \right) }$$.
All formulas on this page are valid under the condition that $$a$$, $$b$$, and $$a+b+2\,n$$ 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 .

7. Parameters[-]

The Jacobi function $${P_{n}^{(a,b)}\left(x\right)}$$ depends on the parameters $$n$$, $$a$$, and $$b$$. The boxes below can be used to rename or instantiate these parameters.
p1 =  p2 =  p3 =