I have a question about eigenvalue problem on the geodesic ball in $n$-dimensional sphere $\mathbb{S}^n\subset\mathbb{R}^{n+1}$.

Consider the eigenvalue problem in the geodesic ball $\Omega=\{x_{n+1}\geq c\}$ where $c\geq 0$:

$\Delta u+nu=0$ in $\Omega$

$u=0$ on $\partial\Omega$.

For the upper hemisphere, i.e. when $c=0$ and $\Omega=\{x_{n+1}\geq 0\}$, the first eigenfunction is given by $u=x_{n+1}$. My question is: for $c>0$, can we find the first eigenfunction explicitly? Can we write down the expression of the first eigenfunction, say, in terms of the coordinate functions? I was told it is related to Legendre function. But I am still not sure how to write down the expression.