The Plancherel–Rotach Formula for Chebyshev–Hermite Functions on Half-Intervals Contracting to Infinity

In this paper, we prove the Plancherel–Rotach asymptotic formula for the Chebyshev–Hermite functions $(-1)^ne^{x^2/2}(e^{-x^2})^{(n)}/\sqrt {2^nn!\sqrt \pi}$ and their derivatives for the case in which $+\infty$ belongs to the domain of definition. A method for calculating the approximation accuracy is also given.