Error bounds for the asymptotic expansions of the Hermite polynomials

In this paper, we present explicit and computable error bounds for the asymptotic expansions of Hermite polynomials with Plancherel-Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory interval, or it is the turning point, are considered respectively. We introduce the branch cut technique to express the error term as an integral on the contour taking as the one-sided limit of curves approaching the branch cut. This new technique enables us to derive simple formulas for the error bounds in terms of elementary functions.