A note on the asymptotic series associated with a certain Titchmarsh-Weyl \(m\)-function (Q1178519)

Let \(m\) be the Titchmarsh-Weyl function of the linear differential equation \(-y''(x)+x^ 2y(x)=\lambda y(x)\) on \([0,\infty)\). The author deals with the asymptotic behaviour of \(m\) in a sector of the form \(0<\varepsilon<\arg\lambda<\pi-\varepsilon\) as \(|\lambda|\to\infty\). The function \(m\) has the asymptotic expansion \(\sum^ \infty_{n=0}i2^{-4n-1}C_{4n+1}\lambda^{- (4n+1)/2}\) where the \(C_ k\) are integers such that \(C_{4m+1}\) is divisible by all prime numbers \(p\) with \(2m+1<p\leq 4m-1\).