# Integral of Hermite functions

In the treatment of the quantum harmonic oscillator appear integrals like $$\begin{equation*} \int_{-\infty}^{+\infty} \mathrm d \zeta \; e^{-\zeta^2} H_{n}(\zeta+\zeta_1) H_{m}(\zeta+\zeta_2) \end{equation*}$$ where $$H_n$$ is the $$n$$-th Hermite function. The expression above is equal to $$\sqrt{\pi}(2^{n} n!)L_{n}(-2\zeta_1\zeta_2)$$ for $$n=m$$, $$L_n$$ being the $$n$$-th Laguerre polynomial. Is there an analogue result for $$n\neq m$$? I am in particularly interested in the cases $$m=n\pm 1$$.

• Are you sure you are not seeking Groenewold's off-diagonal Wigner functions, namely eqn 99 here? – Cosmas Zachos Nov 30 '19 at 22:03
• Groenewold 5.16. Note he slips and calls Laguerres Legendres (!). – Cosmas Zachos Nov 30 '19 at 22:06
• What is that $l$ subscript doing on the $n$ subscript on the $L$? – G. Smith Dec 1 '19 at 0:53
• It is often easier to perform the integrals of this kind on the generating functions of the polynomials and then extract the result you are looking for from the result of the integral. Just remember to give the two generating functions different generating parameters. – flippiefanus Dec 1 '19 at 4:45
• Sorry, thesubscript $l$ was a typo from my side – Graz Dec 1 '19 at 9:18

## 2 Answers

By experimenting in Mathematica I found that

$$\int_{-\infty}^\infty d\zeta e^{-\zeta^2}H_n(\zeta+\zeta_1)H_{n-1}(\zeta+\zeta_2)= \sqrt{\pi}2^n(n-1)!\zeta_1L_{n-1}^1(-2\zeta_1\zeta_2)$$

holds for $$n=1,...,6$$ so I conjecture that it holds for higher values of $$n$$ as well. I didn't try to prove it. It seemed reasonable to guess that if Laguerre polynomials were involved for $$n=m$$, associated Laguerre polynomials might be involved for $$n\ne m$$.

I found in the work of Groenwold, 1946, a closed form for the solution of $$$$k_{mn}(\zeta_1,\zeta_2)=\frac{1}{\sqrt{\pi 2^{m+n} m!n!}}\int_{\mathbb R} \mathrm d \zeta \; e^{-\zeta^2} H_{m}(\zeta+\zeta_1) H_{n}(\zeta+\zeta_2)$$$$ can be written in terms of the associated Laguerre polynomials by considering $$\begin{equation*} \sum_{mn=0}^{+\infty}\sqrt{\frac{2^{m+n}}{m!n!}}k_{mn}(\zeta_1, \zeta_2) \end{equation*}$$ These series can be computed in a closed form by integrating the Gaussians appearing from the expansion of the Hermite polynomials in terms of their generatrix functions. The final solution is then given in terms of the associated Laguerre polynomials, converging to standard Laguerre polynomials for $$m=n$$.