Some new closed expressions for integrals involving Hermite functions (Q1206991)

The author evaluates the integrals \(I_{n,m}= \int_{-\infty}^ \infty e^{-x^ 2} H_ n(x) H_ m(x){{dx} \over x}\), where \(H_ k(\cdot)\) denotes the Hermite polynomial of order \(k\), and related integrals. These quantities are of interest in quantum physics. Based on the recurrence relation for the Hermite polynomials recurrence formulas for \(I_{n,m}\) are obtained and therefrom one gets e.g. that \(I_{n,n+1}=2^{n+1} n! \sqrt{\pi}\), for even \(n\in\mathbb{N}_ 0\).