Uniform bounds for generalized Laguerre polynomials and weighted Cauchy transforms

Prove, for every \(\rho>0\) and \(\alpha>-1\), the conjectured uniform root-exponential bounds on the parabola \(P_\rho\) for the generalized Laguerre polynomials \(L_n^{(\alpha)}\), their weighted Cauchy transforms \(\Phi_n^{(\alpha)}\), and the ratio involving \(\Phi_n^{(\alpha)}/L_n^{(\alpha)}\), including the stronger bounds uniform over all \(n\geq n_0\) for every \(n_0\in\mathbb N_0\).

Background

The authors identify the missing step in the cited Laguerre convergence proofs as justification of convergence under the contour integral. They propose using the dominated convergence theorem, which requires uniform-in-n bounds for the relevant integrands on the unbounded parabola $P_\rho$.

The conjecture asserts three such bounds: one for the weighted Cauchy transform $\Phi_n^{(\alpha)}$, one for the generalized Laguerre polynomial $L_n^{(\alpha)}$, and one for their ratio. A stronger formulation additionally inserts powers of $|z|$ depending on an arbitrary lower index $n_0$, thereby controlling the outer supremum over all $n\geq n_0$. Establishing these estimates would supply the domination needed to complete the convergence proofs.