Let $h_n(x)=(-1)^n\gamma_ne^{x^2/2} \frac{d^n}{dx^n}e^{-x^2}$ where $\gamma_n=\pi^{-1/4}2^{-n/2}(n!)^{-1/2}$ be a Hermite function.
Consider $k_n(x,y)=\frac{h_{n+1}(x)h_n(y)-h_{n+1}(y)h_n(x)}{y-x}$ such that there exists a positive constant $\Gamma$ such that for any $n$ and for all $x$ $\int_{-\infty}^{-1}(k_n(x,y))^2dy< \frac{2\Gamma^2}{n+1}, \quad \int_{1}^{\infty}(k_n(x,y))^2dy< \frac{2\Gamma^2}{n+1}$ I am wondering if one can tell something about constant $\Gamma$, i.e. how small it can be?
Thank you for your help.