Implicit ODE characterization of the eigenvalue curve

Derive an implicit characterization of the curve \(k\mapsto\lambda_{k,\alpha k}\), possibly by identifying an ordinary differential equation whose solution determines the first radial \((k,\alpha k)\)-eigenvalues.

Background

The paper determines the asymptotic limit of λk,n\lambda_{k,n} when n/kn/k tends to a fixed ratio and proves monotonicity for the normalized eigenvalues. It does not provide an implicit description of the full finite-kk curve k↦λk,αkk\mapsto\lambda_{k,\alpha k}. The authors explicitly raise the possibility of obtaining such a description through another ODE.

References

Finally, it is additionally interesting to ask if one can determine the curve $k \mapsto \lambda_{k,\alpha k}$ implicitly, perhaps as the solution to another ODE.

— Asymptotics for the $k$-Hessian Eigenvalue on the Unit Ball  (2609.05277 - McCleerey et al., 4 Sep 2026) in Section 7, Numerical Results and Further Questions

The asymptotic series, \log\consta(p) \simeq 2p2\log p -b p2 + \gamma p\log p + \delta p + \log p +\mu + \dots, likely expands w.r.t. negative powers of the new variable \log p as p\to +\infty. Can their coefficients be constrained by inequalities or algebraic equations\,? In the limit 1/p \to +0, does the asymptotic curve of \consta(p) satisfy any differential equation --- stemming from the expansion of the Wronskian determinant of size 2(p+1) \times 2(p+1) near \consta(p+1) in Eq.~eq:main, that expansion containing the smaller Wronskian determinant, of size 2p\times 2p, as the upper-left minor but itself appearing near~\consta(p)?\

eq:main:

$\mathcal{A}_p[w_1,\ldots,w_N](f)(x) = \consta(p) \cdot \Wronsk \bigl(w_1, \ldots, w_N\bigr)(x) \cdot \partial_x^p(f(x)), $

— New integer sequence OEIS A392714 counts Wronskians: fast evaluation via late-growing permutations  (2610.08636 - Shah et al., 6 Oct 2026) in Section 6, “Open problems and perspectives,” Open problem labeled OpenAsymptSeries