1. Differential Equation[-]

The function $${C_{n}^{(\lambda)}\left(x\right)}$$ satisfies the differential equation$$\displaystyle \left( -{x}^{2}+1 \right) {\frac {d^{2}}{d{x}^{2}}}y \left( x \right) - \left( 2\,\lambda+1 \right) x{\frac {d}{dx}}y \left( x \right) +n \left( n+2\,\lambda \right) y \left( x \right) = 0$$ with initial values $$y \left( 0 \right) =\frac{{2}^{n}\sqrt {\pi }\,\Gamma \left( \lambda+1/2\,n \right) }{\Gamma \left( \lambda \right) \,\Gamma \left( n+1 \right) \,\Gamma \left( 1/2-1/2\,n \right) }$$ and $$y' \left( 0 \right) =-\frac{{2}^{n+1}\sqrt {\pi }\,\Gamma \left( \lambda+1/2\,n+1/2 \right) }{\Gamma \left( n+1 \right) \,\Gamma \left( \lambda \right) \,\Gamma \left( -1/2\,n \right) }$$.
All formulas on this page are valid under the condition that $$2\,\lambda+2\,n$$ and $$\lambda-1/2$$ 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 Gegenbauer ultraspherical function $${C_{n}^{(\lambda)}\left(x\right)}$$ depends on the parameters $$n$$ and $$\lambda$$. The boxes below can be used to rename or instantiate these parameters.
p1 =  p2 =