with $$\sum_{k} |A_{k}(n)|^2 = 1 .$$ This coefficient is $$\label{coeff55}
A_{k}(n)  = \int \chi_{k + n}(x)\chi_{k}(y)\chi_{n}(x')\chi_{0}(y')~ dx~dy .$$ This calculation was given in the literature in a fragmentary way in connection with a Lorentz-covariant description of extended particles starting from Ruiz’s 1974 paper [@ruiz74], subsequently by Kim [*et al.*]{} in 1979 [@kno79ajp] and by Rotbart in 1981 [@rotbart81]. In view of the recent developments of physics, it seems necessary to give one coherent calculation of the coefficient of Eq.(\[coeff55\]).

We are now interested in the squeezed oscillator function $$\begin{aligned}
\label{coef55}
&{}& A_{k}(n) =
\left[\frac{1}{\pi^2 ~2^n n! (k + n)^2 (n + k)! k^2 k!}\right]^{1/2} \nonumber \\[1ex]
&{}& \times \int    H_{n+k}(x) H_{k}(y) H_{n}(x')
\exp{\left\{-\left(\frac{x^2 + y^2 + x'^2 + y'^2 }{2}\right)\right\}}dx dy .\end{aligned}$$ As was noted by Ruiz [@ruiz74], the key to the evaluation of this integral is to introduce the generating function for the Hermite polynomials [@magnus66; @doman16]: $$G(r,z) = \exp{\left(-r^2 + 2rz\right)} = \sum_{m} \frac{r^m}{m!} H_m(z) ,$$ and evaluate the integral $$I = \int G(r,x) G(s,y) G(r',x')
  \exp{\left\{-\left(\frac{x^2 + y^2 + x'^2 + y'^2 }{2}\right)\right\}}dx dy .$$ The integrand becomes one exponential function, and its exponent is quadratic in $x$ and $y$. This quadratic form can be diagonalized, and 