Editing Hermite polynomials (section)

===Wigner distributions of Hermite functions===
The [[Wigner distribution function]] of the {{mvar|n}}th-order Hermite function is related to the {{mvar|n}}th-order [[Laguerre polynomial]]. The Laguerre polynomials are 
<math display="block">L_n(x) := \sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{k!}x^k,</math>
leading to the oscillator Laguerre functions
<math display="block">l_n (x) := e^{-\frac{x}{2}} L_n(x).</math>
For all natural integers {{mvar|n}}, it is straightforward to see<ref>{{Citation |author-link=Gerald Folland |first=G. B. |last=Folland |title=Harmonic Analysis in Phase Space | series=Annals of Mathematics Studies |volume=122 |publisher=Princeton University Press |date=1989 |isbn=978-0-691-08528-9}}</ref> that
<math display="block">W_{\psi_n}(t,f) = (-1)^n l_n \big(4\pi (t^2 + f^2) \big),</math>
where the Wigner distribution of a function {{math|''x'' ∈ ''L''<sup>2</sup>('''R''', '''C''')}} is defined as
<math display="block"> W_x(t,f) = \int_{-\infty}^\infty x\left(t + \frac{\tau}{2}\right) \, x\left(t - \frac{\tau}{2}\right)^* \, e^{-2\pi i\tau f} \,d\tau.</math>
This is a fundamental result for the [[quantum harmonic oscillator]] discovered by [[Hilbrand J. Groenewold|Hip Groenewold]] in 1946 in his PhD thesis.<ref name="Groenewold1946">{{cite journal | last1 = Groenewold | first1 = H. J. | year = 1946 | title = On the Principles of elementary quantum mechanics | journal = Physica | volume = 12 | issue = 7| pages = 405–460 | doi = 10.1016/S0031-8914(46)80059-4 | bibcode=1946Phy....12..405G}}</ref> It is the standard paradigm of [[Phase-space formulation#Simple harmonic oscillator|quantum mechanics in phase space]].

There are [[Laguerre function#Relation to Hermite polynomials|further relations]] between the two families of polynomials.