Explicit determination of the diagonal boundary derivative

Determine an explicit formula for the boundary derivative \(u_{n,n}'(1)\) of the first radial \((n,n)\)-eigenfunction, equivalently determining the associated rescaling parameter \(R_n\) through \(R_n^2=u_{n,n}'(1)\).

Background

For the diagonal case $k=n$, the paper recalls a rescaled eigenfunction and a parameter $R_n$ satisfying $R_n^2=u_{n,n}'(1)$. The authors note that proving monotone convergence of the eigenfunctions would yield that $R_n$ decreases in $n$ and would provide sharp bounds, but would not itself give an explicit formula. The explicit determination of $u_{n,n}'(1)$ therefore remains a separate unresolved question.