Inequalities for orthonormal Laguerre polynomials (Q865368)

The following inequality is established: \[ 10^{-8}<k^{1/6}(k+ \alpha+1)^{-1/2} \max_{x\geq 0}M_k^\alpha(x)<1444(\alpha\geq 24), \] where the upper bound holds for \(k\geq 35\), and lower one for \(k\geq 2 \times 10^{10}\). Also \(M^\alpha_k(x)= (L_k^{(\alpha)}(x))^2e^{-x}x^{\alpha+1}\) and \(L_k^{(\alpha)}(x)\) is an orthonormal polynomial of degree \(k\). Sharp pointwise estimates on \(M_k^\alpha\) and related functions for \(x\geq 0\) are also given.