Equioscillatory property of the Laguerre polynomials (Q619052)

\(L^{(\alpha)}_k\) is the orthonormal Laguerre polynomial of degree \(k\) and \(d_m\), \(d_M\) are some approximations for the extreme zeros. The authors show that the function \[ ((x- d_m)(x- d_M))^{1/4} x^{\alpha/2} e^{-{x\over 2}}L^{(\alpha)}_k(x) \] is almost equi-oscilating with the amplitude \(\sqrt{{2\over\pi}}\) provided \(k\) and \(\alpha\) are large enough. As a corollary they obtain a very explicit, uniform in \(k\) and \(\alpha\), sharp upper bound on Laguerre polynomials.