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\begin{center}
{   \bf POSITION AND MOMENTUM INFORMATION ENTROPIES OF THE
HARMONIC OSCILLATOR AND LOGARITHMIC POTENTIAL OF HERMITE POLYNOMIALS}
\end{center} 
\begin{center}
{\bf Jorge S\'anchez-Ruiz} 
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\begin{abstract}

The exact analytical values of the position and momentum information
entropies for the stationary states of the one-dimensional quantum 
harmonic oscillator are only known for the ground ($n=0$) and first 
excited ($n=1$) states. In the general case, the problem of 
calculating these entropies reduces to the evaluation of the 
logarithmic potential $V_n(t) = - \int_{-\infty}^{\infty} (H_n(x))^2 
\ln \vert x-t \vert e^{-x^2} dx$ at $t=x_{n,i}$ $(i=1,\ldots,n)$, 
where $x_{n,i}$ is the $i$-th zero of the Hermite polynomial $H_n(x)$. 
Here, a closed analytical formula for $V_n(t)$ in terms of $_1F_1$ and 
$_2F_2$ hypergeometric functions is obtained, which in turn provides 
us with analytical expressions for the entropies when the exact 
location of the zeros of $H_n(x)$ is known. The complete asymptotic 
expansion of $V_n(t)$ for $\vert t \vert \gg 1$, which is expressed in 
terms of $_2F_0$ and $_3F_1$ (divergent) hypergeometric series, is 
also obtained. Finally, it is shown that the exact formula for 
$V_n(t)$ can be written as an infinite series involving the Gauss 
(${_2F_1}$) hypergeometric function, which allows the entropies to be 
expressed in terms of the even-order spectral moments of Hermite 
polynomials, $\mu_{2k}(n)= n^{-1} \sum_{i=1}^n (x_{n,i})^{2k}$. 
The asymptotic ($n \rightarrow \infty$) limit of this alternative 
expression for the entropies is discussed.

\end{abstract}



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\section{Introduction}

For the $n$-th eigenstate of the one-dimensional harmonic oscillator 
Hamiltonian, 
\begin{equation}
H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2 \;,
\end{equation}
the probability densities for position and momentum are expressed in 
terms of the orthogonal Hermite polynomials $H_n(x)$,
\begin{equation}
\rho(x) = \frac{\alpha}{2^n n! \sqrt \pi} \big( H_n (\alpha x) 
\big)^2 e^{- \alpha^2 x^2} \;, \quad
\gamma(p) = \frac{1}{2^n n! \sqrt \pi \alpha} \big( H_n ( p/ \alpha )
\big)^2 e^{- p^2 / \alpha^2} \;,
\end{equation}
where $\alpha \equiv \left( m \omega \right)^{1/2}$ (we choose units 
such that $\hbar=1$). The corresponding Boltzmann-Shannon entropies 
of position and momentum \cite{dvaypp,davayti,vayd95,yvad94} can be 
written as
\begin{equation}
S_{\rho} = - \ln \alpha + S_n \;, \quad S_{\gamma} = \ln \alpha + S_n 
\;,
\end{equation}
where
\begin{equation} \label{sss}
S_n = \ln \left( 2^n n! \sqrt{\pi} \; \right) + n + \frac{1}{2} + 
\frac{1}{2^n n! \sqrt{\pi}} \, E_n(H) \;,
\end{equation}
is given in terms of $E_n(H)$, the so-called entropy of Hermite polynomials, whose 
expression is
\begin{equation} \label{sph}
E_n(H) = - \int_{-\infty}^{\infty} e^{-x^2} \big( H_n(x) \big)^2 
\ln \big( H_n(x) \big)^2 \, dx \;.
\end{equation}

The values of $S_n$ have been numerically calculated up to $n=500$ 
\cite{vayd95}. For $n \gg 1$, the $L^p$-norm-based approach to the 
calculation of the entropies of orthogonal polynomials leads to the 
asymptotic formula \cite{abd94,davayti,vayd95} 
\begin{equation} \label{akaf}
S_n \sim \ln \left( \pi \sqrt{2n} \; \right) -1 \;,
\end{equation}
which can also be derived from the semiclassical (WKB) approximation 
for general one-dimensional quantum systems \cite{sr96}. However, the 
exact analytical value of $S_n$ is only known in the simplest cases 
$n=0$ and $n=1$ \cite{dvaypp,davayti,vayd95,yvad94}, 
\begin{equation} \label{n0y1}
S_0 = \ln \left( \sqrt \pi \; \right) + \frac{1}{2} \;, \quad 
S_1 = \ln \left( 2 \sqrt \pi \; \right) - \frac{1}{2} + \gamma \;,
\end{equation}
where $\gamma$ is Euler's constant. The main aim of the present work 
is to find the generalization of these results to arbitrary values of 
$n$ by applying in the Hermite case the potential theory approach to 
the calculation of the entropies, discussed in Ref. \cite{davayti} 
for general orthogonal polynomials.

The Hermite polynomial $H_n(x)$ has $n$ real and simple zeros, and is 
of the form $H_n(x) = 2^n x^n + O(x^{n-1})$ (see, e.g., Ref. 
\cite{gra}), so that it can be factorized as         
\[
H_n(x) = 2^n \prod_{i=1}^n (x-x_{n,i}) \;, 
\]
where $x_{n,i}$ ($i=1,2,\ldots,n$) is the $i$-th root of $H_n(x)$. 
Introducing this expression into the logarithmic function in 
(\ref{sph}), and taking into account the normalization integral for 
Hermite polynomials,
\[
\int_{-\infty}^{\infty} e^{-x^2} \big( H_n(x) \big)^2 \, dx =
2^n n! \sqrt{\pi} \;,
\]
we see that $E_n(H)$ can be written in the form
\begin{equation} \label{sph3}
E_n(H) = - 2^n n! \sqrt{\pi} \, \ln \left( 2^{2n} \right) + 
2 \sum_{i=1}^n V_n(x_{n,i}) \;, 
\end{equation}
where $V_n(t)$ is the logarithmic potential of Hermite polynomials,
\begin{equation} \label{vnt}
V_n(t) = - \int_{-\infty}^{\infty} \big( H_n(x) \big)^2 \ln 
\vert x-t \vert \, e^{-x^2} \, dx \;.
\end{equation}

The $n$ (real and simple) zeros of $H_n(x)$ are symmetrically 
distributed around the origin, since $H_n(-x)=(-1)^n H_n(x)$ 
\cite{gra}. Therefore, it is readily seen that $V_n(-t)=V_n(t)$, and 
Eq. (\ref{sph3}) can also be written as
\begin{equation} \label{sph3otra}
E_n(H) = - 2^n n! \sqrt{\pi} \, \ln \left( 2^{2n} \right) + 
2 \epsilon V_n(0) + 4 \sum_{i=1}^m V_n(x_{n,i}) \;, 
\end{equation}
where we have introduced the convenient notations
\begin{equation} \label{mepdef}
m \equiv \left[ \frac{n}{2} \right], \quad \epsilon \equiv n-2m
\end{equation}
(i.e., $\epsilon$ is equal to 0 or 1 according to whether $n$ is even
or odd), and $x_{n,i}$ ($i=1,2,\ldots,m$) denotes the $i$-th 
{\em positive} root of $H_n(x)$.

Eqs. (\ref{sph3}) and (\ref{sph3otra}) show that the problem of 
calculating $E_n(H)$, and hence $S_n$, reduces to the evaluation of 
$V_n(t)$ at the zeros of $H_n(x)$. In Section 2 below we obtain a 
closed analytical expression for $V_n(t)$ in terms of $_1F_1$ and 
$_2F_2$ hypergeometric functions, which, unlike the recursive formula 
previously derived by Dehesa and co-workers \cite{dvaypp,davayti}, 
provides us with analytical expressions for $E_n(H)$ and $S_n$ when 
the exact location of the zeros of $H_n(x)$ is known. In Section 3, we 
make use of the general theory of asymptotic approximations for the 
$_pF_p$ hypergeometric functions to obtain the complete asymptotic 
expansion of $V_n(t)$ when $\vert t \vert \gg 1$, which is expressed 
in terms of $_2F_0$ and $_3F_1$ (divergent) hypergeometric series. 
Finally, in Section 4, it is shown that the exact formula for $V_n(t)$ 
can be written as an infinite series involving the Gauss ($_2F_1$) 
hypergeometric function, which allows $E_n(H)$ and $S_n$ to be 
expressed in terms of the even-order spectral moments of Hermite 
polynomials $\mu_{2k}(n)$, and this alternative expression for $S_n$ 
is used to discuss the relation between our exact results and the 
asymptotic formula (\ref{akaf}). 

\section{Calculation of the logarithmic potential}

To calculate $V_n(t)$, we first make use of the multiplication formula 
for Hermite polynomials (see, e.g., Ref. \cite{kl93}),          
\[
H_m(x) H_n(x) = \sum_{j=0}^{\min(m,n)} \frac{m! \, n! \, 2^j}
{(m-j)! \, (n-j)! \, j!} H_{m+n-2j} (x) \;,
\]
which in the particular case $m=n$ gives
\[
\big( H_n(x) \big)^2 = \sum_{j=0}^n 
\frac{\big( n! \big)^2 \, 2^j}{\big( (n-j)! \big)^2 j!} H_{2n-2j}(x) = 
2^n \, n! \sum_{k=0}^n \left( \begin{array}{c} n \\ k \end{array} 
\right) \frac{H_{2k}(x)}{2^k k!} \;.
\]
Substituting this equation in the expression (\ref{vnt}) of the 
logarithmic potential $V_n(t)$, we find 
\begin{equation} \label{w2ktdef}
V_n (t) = 2^n n! \sum_{k=0}^n 
\left( \begin{array}{c} n \\ k \end{array} \right) 
\frac{W_{2k}(t)}{2^k k!} \;, \quad
W_{2k} (t) = - \int_{-\infty}^{\infty} H_{2k} (x) \ln \vert x-t \vert 
\, e^{-x^2} \, dx \;.
\end{equation}

Now we are faced with the problem of calculating the integrals 
$W_{2k}(t)$, which can also be considered as logarithmic potentials 
for Hermite polynomials and thus have independent interest. To achieve 
this goal, we consider the Taylor series expansion 
\begin{equation} \label{w2kttse}
W_{2k} (t) = \sum_{r=0}^{\infty} \frac{W_{2k}^{(r)}(0)}{r!} t^r \;. 
\end{equation}
Making the change of variables $x=y+t$ in Eq. (\ref{w2ktdef}), we have
\[
W_{2k} (t) = - \int_{-\infty}^{\infty} H_{2k} (y+t) \ln \vert y \vert 
\, e^{-(y+t)^2} \, dy \;.
\]
By repeated application of Leibniz's rule for differentiating under 
the integral sign, and taking into account that, from Rodrigues'  
formula for Hermite polynomials,
\[
\frac{d^r}{dz^r} \left( e^{-z^2} H_n(z) \right) = 
(-1)^r e^{-z^2} H_{n+r}(z) \;,
\]
we get
\[
W_{2k}^{(r)}(t) = (-1)^r W_{2k+r}(t) \;,
\]
so that Eq. (\ref{w2kttse}) then reads
\begin{equation} \label{w2kttse2}
W_{2k}(t) = \sum_{r=0}^{\infty} \frac{(-1)^r W_{2k+r}(0)}{r!} t^r 
\;. 
\end{equation}
The parity property $H_n(-x) = (-1)^n H_n(x)$ implies that
$W_{2k+r}(0)=0$ if $r$ is odd, and Eq. (\ref{w2kttse2}) simplifies to
\begin{equation} \label{w2kttse3}
W_{2k} (t) = \sum_{r=0}^{\infty} \frac{W_{2k+2r}(0)}{(2r)!} t^{2r} \;, 
\quad W_{2k+2r} (0) = - 2 \int_0^{\infty} H_{2k+2r}(x) \, e^{-x^2} 
\ln x \, dx \;.
\end{equation}

The integrals $W_{2k+2r} (0)$ may be evaluated by means of the 
following result \cite{gra},
\begin{eqnarray*}
\int_0^{\infty} H_{2n}(x) \, x^{\nu} e^{-2 \alpha x^2} \, dx & = & 
(-1)^n \, \frac{2^{ 2n - ( \nu +3)/2}}{\sqrt{\pi} \,
{\alpha}^{( \nu +1)/2}} \; \Gamma \left( \frac{\nu +1}{2} \right) 
\Gamma \left( n+ \frac{1}{2} \right) \nonumber \\ 
& & \times \; F \left( -n, \frac{\nu +1}{2}; \frac{1}{2}; 
\frac{1}{2 \alpha} \right),
\end{eqnarray*}
where $F(a,b;c;z) = {_2F_1} (a,b;c;z)$ is the Gauss hypergeometric 
function, which is valid for ${\rm Re} \: \alpha >0, \; {\rm Re} \: 
\nu > -1$. In our case, with $\alpha= \frac{1}{2}$ and $n=k+r$, we get
\begin{eqnarray} \label{intnu1}
\int_0^{\infty} H_{2k+2r}(x) \, x^{\nu} e^{-x^2} \, dx & = & 
(-1)^{k+r} \, \frac{2^{2k +2r-1}}{\sqrt{\pi}} \; \Gamma \left( 
\frac{\nu +1}{2} \right) \Gamma \left( k+r + \frac{1}{2} \right) 
\nonumber \\ 
& & \times \; F \left( -k-r, \frac{\nu +1}{2}; \frac{1}{2}; 1 \right).
\end{eqnarray}
The hypergeometric function of unit argument in the right-hand side 
can be simplified using the property \cite{luk},
\[
F(-n,b;c;1) = \frac{(c-b)_n}{(c)_n} \;,
\]
where $n$ is a positive integer or zero, $c$ is not a negative integer 
or zero, and $(z)_n$ is Pochhammer's symbol,
\begin{equation} \label{pochdef}
(z)_n = \frac{\Gamma(z+n)}{\Gamma(z)} = (-1)^n \frac{\Gamma(1-z)}
{\Gamma(1-z-n)} \;.
\end{equation}
We thus have
\[
F \left( -k-r, \frac{\nu +1}{2}; \frac{1}{2}; 1 \right) = 
\frac{\left( - \frac{\nu}{2} \right)_{k+r}}{\left( \frac{1}{2} 
\right)_{k+r}} = \frac{\Gamma \left( \frac{1}{2} \right) 
\Gamma \left( k+r- \frac{\nu}{2} \right)}
{\Gamma \left( k+r+ \frac{1}{2} \right) \Gamma \left( - 
\frac{\nu}{2} \right)} \;,
\]
and Eq. (\ref{intnu1}) then reads
\[
\int_0^{\infty} H_{2k+2r}(x) \, x^{\nu} e^{-x^2} \, dx = (-1)^{k+r} \,
2^{2k +2r-1} \: \Gamma \left( \frac{\nu +1}{2} \right) 
\frac{ \Gamma \left( k+r- \frac{\nu}{2} \right)}{ \Gamma \left( 
- \frac{\nu}{2} \right)} \;.
\]
Differentiating this equation with respect to $\nu$, we obtain
\begin{eqnarray} \label{intnuln}
\int_0^{\infty} H_{2k+2r}(x) \, x^{\nu} e^{-x^2} \ln x \, dx & = &
(-1)^{k+r} \, 2^{2k +2r-1} \: \Gamma \left( \frac{\nu +1}{2} \right) 
\frac{ \Gamma \left( k+r- \frac{\nu}{2} \right)}
{ \Gamma \left( - \frac{\nu}{2} \right)} \nonumber \\
& & \times \frac{1}{2} \left( \psi \left( \frac{\nu +1}{2} \right) 
\! - \psi \left( k+r- \frac{\nu}{2} \right) 
\! + \psi \left( \! - \frac{\nu}{2} \right) \! \right) \!, 
\end{eqnarray}
where $\psi(z)=\Gamma^{\prime}(z) / \Gamma(z)$ is the logarithmic
derivative of the Gamma function. In the case when $k+r=0$, this 
formula reduces to
\[
\int_0^{\infty} H_0(x) \, x^{\nu} e^{-x^2} \ln x \, dx = 
\frac{1}{4} \, \Gamma \left( \frac{\nu +1}{2} \right) 
\psi \left( \frac{\nu +1}{2} \right),
\]
so that we readily get
\begin{equation} \label{w00}
W_0 (0) = - \frac{1}{2} \, \Gamma \left( \frac{1}{2} \right) 
\psi \left( \frac{1}{2} \right) = 
\frac{\sqrt{\pi}}{2} \left( \gamma + 2 \ln 2 \right) \;.
\end{equation}
On the other hand, both $\Gamma(z)$ and $\psi(z)$ have simple poles 
for $z=0$, with residues $1$ and $-1$ respectively. Therefore, when 
$k+r>0$, we can take the limit $\nu \rightarrow 0$ in Eq. 
(\ref{intnuln}), to obtain
\[
\int_0^{\infty} H_{2k+2r}(x) \, e^{-x^2} \ln x \, dx =
- \frac{\sqrt{\pi}}{4} \, (-1)^{k+r} \, 2^{2k+2r} \, \Gamma (k+r) \;, 
\quad k+r > 0 \;,
\]
which in turn leads to
\begin{equation} \label{wp0}
W_{2k+2r} (0) = \frac{\sqrt{\pi}}{2} \, (-1)^{k+r} \, 2^{2k+2r} \, 
\Gamma (k+r) \;, \quad k+r > 0 \;.
\end{equation}

We can evaluate $W_{2k}(t)$ by substituting Eqs. (\ref{w00}) and 
(\ref{wp0}) into (\ref{w2kttse3}). In the case $k=0$, we have
\begin{equation} \label{w0tcal}
W_0 (t) = W_0 (0) + \sum_{r=1}^{\infty} \frac{W_{2r}(0)}{(2r)!} t^{2r} 
= \frac{\sqrt \pi}{2} \left( \gamma + 2 \ln 2 + \sum_{r=1}^{\infty} 
\frac{(-1)^r 2^{2r} \Gamma(r)}{\Gamma(2r+1)} t^{2r} \right) .
\end{equation}
Using the recurrence and duplication formulas for the Gamma function
\cite{gra,luk},
\[
\Gamma (z+1) = z \Gamma (z) \;, \quad \Gamma (2z) = 
2^{2z-1} \left( \frac{1}{2} \right)_{\! z} \Gamma (z) \;,
\]
together with (\ref{pochdef}), and shifting the summation index to 
$s=r-1$, Eq. (\ref{w0tcal}) can be written in terms of a ${_2F_2}$ 
hypergeometric function,
\begin{equation} \label{w0t}
W_0 (t) = \frac{\sqrt{\pi}}{2} \left( \gamma + 2 \ln 2 - 2 {t^2} \,
{_2F_2 \left( 1,1; \frac{3}{2}, 2; -t^2 \right)} \right) .
\end{equation}
On the other hand, for $k>0$ we have
\begin{equation} \label{w2ktcal}
W_{2k} (t) = \sum_{r=0}^{\infty} \frac{W_{2k+2r}(0)}{(2r)!} t^{2r} =
\frac{\sqrt{\pi}}{2} (-1)^k 2^{2k} \sum_{r=0}^{\infty} 
\frac{(-1)^r 2^{2r} \Gamma (k+r)}{\Gamma (2r+1)} t^{2r} \;,
\end{equation}
and use of the duplication formula for the Gamma function and 
Eq. (\ref{pochdef}) leads to
\begin{equation} \label{w2kt}
W_{2k} (t) = \frac{\sqrt{\pi}}{2} (-1)^k 2^{2k} (k-1)! \,
M \! \left( k; \frac{1}{2}; -t^2 \right) ,
\end{equation}
where $M(a;c;z)={_1F_1(a;c;z)}$ is Kummer's confluent hypergeometric 
function. Substituting Eqs. (\ref{w0t}) and (\ref{w2kt}) into 
(\ref{w2ktdef}), we finally obtain
\begin{eqnarray} \label{vntmr}
V_n(t) & = & 2^n n! \sqrt{\pi} \left( \, \frac{\gamma}{2} + \ln 2 - 
{t^2} \, {_2F_2 \left( 1,1; \frac{3}{2}, 2; -t^2 \right)} \right. 
\nonumber \\ 
& & \left. + \: \frac{1}{2} \, \sum_{k=1}^n \left( \begin{array}{c} 
n \\ k \end{array} \right) \frac{(-1)^k 2^k}{k} \, 
M \! \left( k; \frac{1}{2}; -t^2 \right) \right),
\end{eqnarray}
which is the sought for closed analytical expression for the 
logarithmic potential $V_n(t)$ defined by Eq. (\ref{vnt}).

In the particular case $t=0$, using the identity \cite{srup}
\begin{equation} \label{sfsbis}
\gamma + 2 \ln 2 + \sum_{k=1}^n \left( \begin{array}{c} n \\ k 
\end{array} \right) \frac{(-1)^k 2^k}{k} = - \, \psi \left( 
m + \epsilon + \frac{1}{2} \, \right),
\end{equation}
where $m$ and $\epsilon$ are defined by Eq. (\ref{mepdef}), 
Eq. (\ref{vntmr}) reduces to
\begin{equation} \label{vnt0}
V_n(0) = - 2^{n-1} n! \sqrt{\pi} \, \psi \left( m + \epsilon 
+ \frac{1}{2} \, \right) .
\end{equation}

A closed formula for the entropy of Hermite polynomials $E_n(H)$ can 
be obtained by combining Eqs. (\ref{sph3}) and (\ref{vntmr}),
\begin{eqnarray} \label{sphmr}
E_n (H) & = & 2^n n! \sqrt{\pi} \left( n \gamma - 2 \sum_{i=1}^n 
x_{n,i}^2 \, _2F_2 \left( 1,1; \frac{3}{2}, 2; -x_{n,i}^2 \right) 
\right. \nonumber \\ 
& & \left. + \sum_{k=1}^n \left( \begin{array}{c} n \\ k \end{array} 
\right) \frac{(-1)^k 2^k}{k} \sum_{i=1}^n M \left( k; \frac{1}{2}; 
-x_{n,i}^2 \right) \right).
\end{eqnarray}
Alternatively, using Eq. (\ref{sph3otra}) instead of (\ref{sph3}), and 
taking into account Eq. (\ref{vnt0}), together with \cite{gra,luk}
\[
\psi \left( n + \frac{1}{2} \right) = - \gamma - 2 \ln 2 + 2
\sum_{k=1}^n \frac{1}{2k-1} \;,
\]
we get
\begin{eqnarray} \label{sphmrotra}
E_n (H) & = & 2^n n! \sqrt{\pi} \left( n \gamma - 2 \epsilon 
\sum_{k=1}^{m+ \epsilon} \frac{1}{2k-1} - 4 \sum_{i=1}^m x_{n,i}^2 \, 
_2F_2 \left( 1,1; \frac{3}{2}, 2; -x_{n,i}^2 \right) \right. 
\nonumber \\
& & \left. + \; 2 \sum_{k=1}^n \left( \begin{array}{c} n \\ k 
\end{array} \right) \frac{(-1)^k 2^k}{k} \sum_{i=1}^m M 
\left( k; \frac{1}{2}; -x_{n,i}^2 \right) \right) .
\end{eqnarray}
In turn, from these results, using Eq. (\ref{sss}) we obtain for $S_n$ 
the expressions
\begin{eqnarray} \label{sn1}
S_n & = & \ln \left( 2^n n! \sqrt{\pi} \; \right) + n + \frac{1}{2} + 
n \gamma - 2 \sum_{i=1}^n x_{n,i}^2 \, _2F_2 
\left( 1,1; \frac{3}{2}, 2; -x_{n,i}^2 \right) \nonumber \\ 
& & + \sum_{k=1}^n \left( \begin{array}{c} n \\ k \end{array} \right) 
\frac{(-1)^k 2^k}{k} \sum_{i=1}^n M \left( k; \frac{1}{2}; -x_{n,i}^2 
\right),
\end{eqnarray}
and 
\begin{eqnarray} \label{sn2}
S_n & = & \ln \left( 2^n n! \sqrt{\pi} \; \right) + n + \frac{1}{2} + 
n \gamma - 2 \epsilon \sum_{k=1}^{m+ \epsilon} \frac{1}{2k-1} - 4 
\sum_{i=1}^m x_{n,i}^2 \, _2F_2 \left( 1,1; \frac{3}{2}, 2; -x_{n,i}^2 
\right) \nonumber \\
& & + \; 2 \sum_{k=1}^n \left( \begin{array}{c} n \\ k \end{array} 
\right) \frac{(-1)^k 2^k}{k} \sum_{i=1}^m M \left( k; \frac{1}{2}; 
-x_{n,i}^2 \right),
\end{eqnarray}
respectively, which are the generalization of Eq. (\ref{n0y1}) to 
arbitrary values of $n$. For example, in the $n=2$ case, the zeros of 
the polynomial $H_2(x) = 4x^2-2$ are $\pm \frac{1}{\sqrt 2}$, so that 
we have
\[ 
S_2 = \ln \left( 8 \sqrt{\pi} \; \right) + \frac{5}{2} + 2 \gamma - 
2 \, {_2F_2 \left( 1,1; \frac{3}{2}, 2; -\frac{1}{2} \right)} -
8 \, M \left( 1; \frac{1}{2}; -\frac{1}{2} \right) +
4 \, M \left( 2; \frac{1}{2}; -\frac{1}{2} \right),
\]
while in the $n=3$ case, the zeros of $H_3(x) = 8 x^3 - 12x$ are 
$0, \pm \sqrt{\frac{3}{2}}$, and we have
\begin{eqnarray*} 
S_3 & = & \ln \left( 48 \sqrt{\pi} \; \right) + \frac{5}{6} + 3 \gamma 
- 6 \, {_2F_2 \left( 1,1; \frac{3}{2}, 2; -\frac{1}{2} \right)} -
12 \, M \left( 1; \frac{1}{2}; -\frac{3}{2} \right) \\
& & + 12 \, M \left( 2; \frac{1}{2}; -\frac{3}{2} \right) - 
\frac{16}{3} \, M \left( 3; \frac{1}{2}; -\frac{3}{2} \right).
\end{eqnarray*}

\section{Asymptotic expansion}

The complete asymptotic expansion of the confluent hypergeometric 
function $M(a,c,z)$ for $z$ real, $\vert z \vert \rightarrow \infty$ 
is given by \cite{luk},
\begin{eqnarray*}
M(a,c,z) & \sim & \frac{\Gamma(c)}{\Gamma(c-a)} {(-z)^{-a}} \, {_2F_0 
\left( a,1+a-c;-z^{-1} \right)} \nonumber \\ 
& & + \: \frac{\Gamma(c)}{\Gamma(a)} e^z {z^{a-c}} \,
{_2F_0 \left( c-a,1-a;z^{-1} \right)}.
\end{eqnarray*}
If $z=-t^2$ is a negative real number, the first term is dominant 
since the second one vanishes exponentially for $z \rightarrow - 
\infty$. Therefore, for $a=k, c=\frac{1}{2}$, we have 
\begin{eqnarray} \label{a1f1}
M \left( k,\frac{1}{2},-t^2 \right) & \sim & \frac{\Gamma \left( 
\frac{1}{2} \right)}{\Gamma \left( \frac{1}{2} -k \right)} {t^{-2k}}
\, {_2F_0 \left( k,k+\frac{1}{2};t^{-2} \right)} \nonumber \\ 
& = & \frac{(2k-1)!}{2^{2k-1}(k-1)!} (-1)^k \, {t^{-2k}}
\, {_2F_0 \left( k,k+\frac{1}{2};t^{-2} \right)} .
\end{eqnarray}

More generally, for $_pF_p$ hypergeometric functions, it has been 
shown \cite{luk} that, for $\vert z \vert \rightarrow \infty$, 
$\arg z =0$,
\[
_pF_p( \alpha_1, \alpha_2, \ldots, \alpha_p; \rho_1, \rho_2, \ldots,
\rho_p; -z) \sim \frac{\prod_{j=1}^p \Gamma(\rho_j)}
{\prod_{j=1}^p \Gamma(\alpha_j)} \,
\big( K_{p,p} (-z) + L_{p,p} (z) \big) \;.
\]
The function $K_{p,q}(z)$ is defined as
\[
K_{p,q}(z) = \frac{(2 \pi)^{(1-\beta)/2}}{{\beta}^{1/2}}
\exp \left( \beta z^{1/ \beta} \right) z^{\delta} 
\sum_{r=0}^{\infty} N_r \, z^{-r/ \beta} \;, 
\]
where
\[
\beta = q+1-p \;, \quad \delta = \frac{1}{\beta} \left( 
\frac{\beta -1}{2} + \sum_{j=1}^p \alpha_j - \sum_{j=1}^q \rho_j 
\right) , \quad N_0 = 1 \;,
\]
and for $r \geq 1$ the constants $N_r$ are determined by means of 
recurrent formulas, while the function $L_{p,q}(z)$ can be formally 
written for arbitrary values of $p$ and $q$ in terms of Meijer's $G$ 
function as
\[ 
L_{p,q}(z) = G_{q+1,p}^{p,1} \left( z^{-1} \left| \begin{array}{c} 
{1, \rho_1, \rho_2, \ldots, \rho_q} \\ {\alpha_1, \alpha_2, \ldots, 
\alpha_p} \end{array} \right. \right) .
\]
In the particular case $p=q=2, \, \alpha_1=\alpha_2=1, \,
\rho_1=\frac{3}{2}, \, \rho_2=2$, we have
\[
_2F_2 \left( 1,1; \frac{3}{2}, 2; -t^2 \right) \sim 
\frac{\sqrt \pi}{2} \, \big( K_{2,2} (-t^2) + L_{2,2} (t^2) \big) \;,
\]
where
\[
K_{2,2}(z) = e^z z^{-3/2} \sum_{r=0}^{\infty} N_r \, z^{-r} \;, \quad 
L_{2,2}(z) = G_{3,2}^{2,1} \left( z^{-1} \left| \begin{array}{c} 
{1,\frac{3}{2},2} \\ {1,1} \end{array} \right. \right) .
\]
We see that $K_{2,2}(-t^2)$ vanishes exponentially for $\vert t \vert 
\rightarrow \infty$. On the other hand, using results in Ref. 
\cite{luk}, it can be shown \cite{srup} that 
\[
G_{3,2}^{2,1} \left( z^{-1} \left| \begin{array}{c} 
{1,\frac{3}{2},2} \\ {1,1} \end{array} \right. \right) =
\frac{1}{\sqrt{\pi} z} \left( \ln \vert z \vert + \gamma + 
2 \ln 2 - \frac{1}{2z} \, {_3F_1} \left( 1,1,\frac{3}{2};2;z^{-1}
\right) \right) ,
\]
so that we get 
\begin{equation} \label{a2f2}
_2F_2 \left( 1,1; \frac{3}{2}, 2; -t^2 \right) \sim 
\frac{1}{2 t^2} \left( 2 \ln \vert t \vert + \gamma + 
2 \ln 2 - \frac{1}{2 t^2} \, {_3F_1} \left( 1,1,\frac{3}{2};2;t^{-2}
\right) \right).
\end{equation}
Substituting Eqs. (\ref{a1f1}) and (\ref{a2f2}) into (\ref{vntmr}),
we finally obtain 
\begin{eqnarray} \label{vntamr}
V_n(t) & \sim & 2^n n! \sqrt{\pi} \left( - \ln \vert t \vert + 
\frac{1}{4t^2} \, {_3F_1} \left( 1,1, \frac{3}{2}; 2; t^{-2} \right) 
\right. \nonumber \\ 
& & \left. + \sum_{k=1}^n \left( \begin{array}{c} n \\ k \end{array} 
\right) \frac{(2k-1)!}{2^k k!} t^{-2k} \, {_2F_0} 
\left( k, k+\frac{1}{2}; t^{-2} \right) \right),
\end{eqnarray}
which is the complete asymptotic expansion of $V_n(t)$ for 
$\vert t \vert \gg 1$.

\section{Alternative expressions}

Using Eq. (\ref{w0tcal}) for $W_0(t)$, and Eq. (\ref{w2ktcal}) for 
$W_{2k}(t)$, $k>0$, Eq. (\ref{vntmr}) can be written as
\begin{eqnarray*}
V_n(t) & = & 2^n n! \sqrt{\pi} \left( \frac{\gamma}{2} + \ln 2 + 
\frac{1}{2} \sum_{r=1}^{\infty} \frac{(-1)^r (r-1)!}{(2r)!} (2t)^{2r} 
\right. \nonumber \\ 
& & \left. + \; \frac{1}{2} \, \sum_{k=1}^n \left( \begin{array}{c} 
n \\ k \end{array} \right) \frac{(-1)^k 2^k}{k!} \sum_{r=0}^{\infty} 
\frac{(-1)^r \Gamma (k+r)}{(2r)!} (2t)^{2r} \right) ,
\end{eqnarray*}
which, taking advantage of Eq. (\ref{sfsbis}), simplifies to
\begin{eqnarray} \label{vntalt1}
V_n(t) & = & 2^{n-1} n! \sqrt{\pi} \left( - \, \psi 
\left( m + \epsilon + \frac{1}{2} \right) + \sum_{r=1}^{\infty} 
\frac{(-1)^r (r-1)!}{(2r)!} (2t)^{2r} \right. \nonumber \\ 
& & \left. + \sum_{k=1}^n \left( \begin{array}{c} n \\ k \end{array} 
\right) \frac{(-1)^k 2^k}{k!} \sum_{r=1}^{\infty} \frac{(-1)^r 
\Gamma(k+r)}{(2r)!} (2t)^{2r} \right) .
\end{eqnarray}
Taking into account Eq. (\ref{pochdef}), together with the identity 
\cite{luk}
\[
\left( \begin{array}{c} n \\ k \end{array} \right) =
\frac{(-1)^k (-n)_k}{k!} \;,
\]
the summation over $k$ in the double series of Eq. (\ref{vntalt1}) can 
be performed in terms of the Gauss hypergeometric function, 
\[
\sum_{k=1}^n \left( \begin{array}{c} n \\ k \end{array} \right) 
\frac{(-1)^k 2^k \Gamma(r+k)}{k!} = (r-1)! \: 
\big( F(r,-n;1;2)-1 \big) \;.
\]
Substituting this result into (\ref{vntalt1}), we get
\begin{equation} \label{mr3}
V_n(t) = 2^{n-1} n! \sqrt \pi \left( - \psi \left( m + \epsilon + 
\frac{1}{2} \right) + \sum_{r=1}^{\infty} \frac{(-1)^r 2^{2r} (r-1)!}
{(2r)!} F(r,-n;1;2) t^{2r} \right) ,
\end{equation}
which is an alternative expression for the logarithmic potential 
$V_n(t)$.

The entropy of the Hermite polynomials defined by Eq. (\ref{sph3}) 
can thus be written in the form
\begin{eqnarray} \label{sph5}
E_n(H) & = & 2^n n! \sqrt{\pi} \left( - \ln \left( 2^{2n} \right) 
- n \: \psi \left( m + \epsilon + \frac{1}{2} \right) \right. 
\nonumber \\ 
& & \left. + \; n \sum_{r=1}^{\infty} \frac{(-1)^r 2^{2r} (r-1)!}
{(2r)!} F(r,-n;1;2) \mu_{2r}(n) \right) ,
\end{eqnarray}
where $\mu_r(n)$ ($r=0,1,2,\ldots$) are the spectral moments around 
the origin of the Hermite polynomial $H_n(x)$, i.e., the quantities
\[
\mu_r(n) = \frac{1}{n} \sum_{i=1}^n \big( x_{n,i} \big)^r \;.
\]
Eq. (\ref{sss}) then yields, taking into account that $m+\epsilon = 
(n+\epsilon)/2$, 
\begin{eqnarray} \label{snalt}
S_n & = & \ln \left( \frac{n! \sqrt \pi}{2^n} \right) +n + \frac{1}{2} 
- n \: \psi \left( \frac{n+\epsilon+1}{2} \right) \nonumber \\ 
& & + \; n \sum_{r=1}^{\infty} \frac{(-1)^r 2^{2r}(r-1)!}{(2r)!} 
F(r,-n;1;2) \mu_{2r}(n) \;.
\end{eqnarray}

This new expression for $S_n$ is less useful than Eqs. (\ref{sn1}) and 
(\ref{sn2}) for practical calculations, since, unfortunately, there is 
no global and compact expression for the moments $\mu_{2r}(n)$, but 
they have to be recurrently generated \cite{c80}. 
However, Eq. (\ref{snalt}) turns out to be more appropriate than Eqs. 
(\ref{sn1}) and (\ref{sn2}) to discuss the problem of the relation 
between our present exact results and the asymptotic approximation 
(\ref{akaf}). Use of the well-known asymptotic expansions for the 
Gamma and Psi functions \cite{luk} gives
\begin{eqnarray*}
& & \ln \: (n!) \sim \left( n + \frac{1}{2} \right) \ln n - n + 
\frac{1}{2} \ln \left( 2 \pi \right) + O (n^{-1}) \;, \nonumber \\
& & \psi \left( \frac{n+\epsilon+1}{2} \right) \sim \ln \left( 
\frac{n+\epsilon}{2} \right) + O(n^{-2}) \sim 
\ln n - \ln 2 + \frac{\epsilon}{n} + O(n^{-2}) \;,
\end{eqnarray*}
where the remaining terms of these expansions can be explicitly 
written in terms of Bernoulli numbers. Substituting these 
expressions into Eq. (\ref{snalt}), we obtain the asymptotic 
formula
\[
S_n \sim \ln \left( \pi \sqrt{2n} \; \right) + \frac{1}{2} - \epsilon 
+n \sum_{r=1}^{\infty} \frac{(-1)^r 2^{2r} (r-1)!}{(2r)!} F(r,-n;1;2) 
\mu_{2r}(n) + O(n^{-1}) \;,
\]
and comparison of this result with Eq. (\ref{akaf}) yields 
\[ 
n \sum_{r=1}^{\infty} \frac{(-1)^r 2^{2r} (r-1)!}{(2r)!} F(r,-n;1;2) 
\mu_{2r}(n) \sim \epsilon - \frac{3}{2} + o(1) \;. 
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{ACKNOWLEDGEMENTS} 

The author gratefully acknowledges the warm hospitality of Professor
J. S. Dehesa at the Universidad de Granada (Spain), where part of this 
reasearch was carried out, and the financial support of the Fundaci\'o 
Aula (Barcelona, Spain).

\vspace*{0.3cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The bibliography   %%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In alphabetical order  %%%%%%%%%%%%%%%%%%%%%%%%

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\bibitem{yvad94} {\sf R. J. Y\'a\~nez, W. Van Assche, J. S. Dehesa}: 
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\end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Your address %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\sc 
\begin{flushleft}
{ Jorge S\'anchez-Ruiz } \\
{ Departament de F\'{\i}sica Fonamental, Universitat de 
Barcelona, \\ Diagonal 647, 08028 Barcelona, Spain }  \\
{\sf E-mail:}{ \tt jsanchez@hermes.ffn.ub.es }  
\end{flushleft} 
 }
\end{document}

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