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\begin{center} 
{ \large \bf SOME NEW MATRIX FORMULAS RELATED TO HERMITE MATRIX POLYNOMIALS 
THEORY} 
\end{center}  
\begin{center} 
{  \bf Lucas J\'odar and Emilio Defez} 
\end{center} 
 
\vspace*{1cm} 
 
\begin{abstract} 
In this paper an integral representation for the Hermite 
matrix polynomials is given. By means of the exact computation of certain 
matrix integrals and the integral representation of Hermite matrix 
polynomials, a formula for the generating function of the product of Hermite 
matrix polynomials is obtained. Both the integral representation of the Hermite 
matrix polynomials and the formula for the generating function of the product 
of the Hermite matrix polynomials coincide in the scalar case with the 
well-known formulas for the classical Hermite polynomials. 
\end{abstract} 
 
\vspace*{1cm} 

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\section{Introduction} 
General orthogonal polynomials play an important role in approximation theory, 
numerical analysis and the analytic solutions of boundary problems for partial 
differential, see for example \cite{da}, \cite{de}, \cite{ha}, \cite{fo}. The 
Hermite polynomials arise in the study of classical boundary value problems 
in parabolic regions, through the use of parabolic coordinates, or in quantum 
mechanics, see \cite[chapter 6]{fo}, \cite{17}, as well as in other application areas. 
 
\vspace{1mm} 
 
Orthogonal polynomials which have matrix coefficients appear in connection  
with representation theory, matrix expansion problems, prediction theory  
\cite{ba}, \cite{ph}, \cite{ge}, \cite{ul} and in the reconstruction of  
matrix functions. Those orthogonal on the 
unit circle are considered in \cite{ro}, and matrix polynomials orthogonal 
on the real line have been treated in \cite{18} and \cite{si}, and, in the  
last years in \cite{si}, \cite{jo}, 
\cite{03}, \cite{04}. In \cite{13}, \cite{14} and \cite{15}, 
Laguerre, Gegenbauer and Hermite matrix polynomials have been introduced. 
 
\vspace{1mm} 
 
If $H_n(x)$ denotes the $nth$ scalar Hermite polynomial, the formulas 
 
\begin{equation}\label{k.1.1} 
H_n(x)=\frac{2^{n}(-i)^{n}e^{x^2}}{\sqrt{\pi}}\int_{-\infty}^{+\infty} 
e^{-t^2+2ixt}t^ndt \ \ , \ \ n=0,1,2,\ldots, 
\end{equation} 
 
and 
 
\begin{equation}\label{k.1.2} 
W(x,y,t)=(1-t^2)^{-\frac{1}{2}}e^{\frac{\left[2xyt-(x^2+y^2)t^2\right]} 
{(1-t^2)}}=\sum_{n \geq 0}\frac{H_n(x)H_n(y)}{2^nn!}t^n \ \ , \ \ |t|<1 \ \ , 
\end{equation} 
 
play a fundamental role in the study of the convergence of the expansions in 
series of Hermite polynomials, as well as in the computation 
of the coefficients of such series, see \cite[chapter 4]{17}. 
 
\vspace{1mm} 
 
In the study of matrix functions and series of Hermite 
matrix polynomials $H_n(x,A)$, it turns out that an integral 
representation of $H_n(x,A)$, as well as the exact evaluation of the matrix 
series 
 
$$ 
\sum_{n \geq 0}\frac{H_n(x,A)H_n(y,A)}{2^nn!}t^n  \ \ ,  
$$ 
are necessary. This motivates the search for matrix analogues of formulas  
(\ref{k.1.1})-(\ref{k.1.2}), and it constitutes the aim of this paper whose  
organization is as follows. In section 2 we obtain an integral representation  
of $\exp\left(-\frac{Ax^2}{2}\right)$ for matrices in $C^{r \times r}$ whose 
eigenvalues lie in the right-half plane, and this is used to give an integral 
representation of the Hermite matrix polynomials. In section 3   
parametric matrix integrals of the type 
 
$$ 
\int_{-\infty}^{+\infty}\exp\left( Bxt \right) 
\exp \left( -\frac{Ax^2}{2} \right)dx 
$$ 
 
are calculated in a closed form. Finally, in section 4, a matrix formula for  
the generating function of the product of Hermite matrix polynomials is 
established. 
 
\vspace{1mm} 
 
Throughout this paper the set of all the eigenvalues of a matrix $D$ in 
$C^{r \times r}$ will be denoted by $\sigma(D)$, and $\|D\|$ will denote the 
$2$-norm of $D$, defined by 
 
$$ 
\|D\|=\sup_{x \neq 0}\frac{\|Dx\|_2}{\|x\|_2} \ \ , 
$$ 
 
where, for a vector $y$ in $C^r$, $\|y\|_2$ denotes the usual euclidean norm of 
$y$. If $D$ belongs to $C^{r \times r}$ and $\alpha(D)=\max\left\{Re z \ \ ; 
\ \ z \in \sigma(D)\right\}$, then by \cite[p.556]{10},  
 
\begin{equation}\label{k.1.4} 
\|e^{xD}\| \leq e^{x\alpha(D)}\sum_{k=0}^{r-1}\frac{\left( 
\|D\|x\sqrt{r} \right)^k}{k!} \ \ , \ \ x \geq 0 \ \ . 
\end{equation} 
 
In what follows, the word integrable means integrable in the Lebesgue sense. 
 
\section{Integral representation of the Hermite matrix polynomials.} 
 
The following result is the starting point of the paper. 
 
\begin{theorem}\label{k.t.2.2} 
Let $A$ be a matrix in $C^{r \times r}$ satisfying the condition 
 
\begin{equation}\label{k.2.1} 
Re \, z \, > \, 0 \ \ \ for \ \ every \ \ z \in \sigma (A) \ \ . 
\end{equation} 
 
Then it follows that 
 
\begin{equation}\label{k.2.2} 
\exp \left( -\frac{Ax^2}{2}\right)=2\left(\frac{2A^{-1}}{\pi}\right)^{ 
\frac{1}{2}}\int_{0}^{\infty}\exp\left(-2v^2A^{-1}\right)\cos{2vx}dv \ \ , 
\ \ -\infty < x < +\infty \ \ . 
\end{equation} 
\end{theorem} 
 
\underline{Proof.} Let us consider the parametric matrix integral 
 
\begin{equation}\label{k.2.3} 
J(x)=\int_{0}^{\infty}\exp\left(-2A^{-1}v^2\right)\cos{2vx}dv \ \ , \ \ 
-\infty \, < \, x \, < \, \infty \ \ , 
\end{equation} 
 
and note that if $z=x+iy$,  
 
\begin{equation}\label{k.2.4} 
\frac{1}{z}=\frac{\bar{z}}{|z|^{2}} \ \ and \ \ thus \ \ 
Re\left(\frac{1}{z}\right)=\frac{Re\left( 
z\right)}{|z|^{2}} \ \ . 
\end{equation} 
 
By the spectral mapping theorem \cite[p.569]{02}, and (\ref{k.2.1}) and  
(\ref{k.2.4}), it follows that 
 
\begin{equation}\label{k.2.5} 
Re \omega >0 \ \ for \ \ every \ \ \omega \in \sigma (A^{-1}) \ \ . 
\end{equation} 
 
From the inequalities (\ref{k.1.4})  and (\ref{k.2.5}), the absolute convergence 
of $J(x)$ defined by (\ref{k.2.3}) for any real number $x$, as well as the uniform 
convergence of the integral 
 
$$ 
\int_{0}^{\infty}\, v \,\exp\left(-2A^{-1}v^2\right)\sin{2vx}dv \ \ , 
$$ 
 
in every neighbourhood of $x$. By the Leibnitz theorem for the  
differentiation of matrix integrals, \cite[p.174]{01}, \cite{20}, and  
integrating by parts, it follows that 
 
\begin{displaymath} 
J'(x) = -xAJ(x)+\frac{A}{2} \left[\exp \left(-2A^{-1} 
            v^2 \right) \sin{2xv} \right]_{v=0}^{v=\infty} \ \ . 
\end{displaymath} 
 
Now, again by (\ref{k.1.4}), one gets 
 
$$ 
\left[\exp \left(-2A^{-1}v^2 \right) \sin{2xv} \right]_{v=0}^{v=\infty}=0 
\ \ , 
$$ 
 
and 
 
$$ 
J'(x)=-xAJ(x) \ \ . 
$$ 
 
Taking $x=0$ in (\ref{k.2.3}), and using the expression (4.8) of \cite{15}, 
 
\begin{displaymath} 
J(0)=\int_{0}^{\infty} \exp \left(-2 A^{-1} v^{2}\right) dv =  
\frac{\left(\pi A \right)^{\frac{1}{2}}}{2\sqrt{2}} \ \ . 
\end{displaymath} 
 
Thus $J(x)$ is a solution of the initial value problem 
 
\begin{equation}\label{k.2.6} 
J'(x)=-xAJ(x) \ \ ; \ \ J(0)=\frac{1}{2}\left(\frac{\pi A}{2}\right)^{\frac 
{1}{2}} \ \ , \ \ -\infty < x < +\infty \ \ . 
\end{equation} 
 
Since $D(x)=xA$ satisfies the condition $D(x_1)D(x_2)=D(x_2)D(x_1)$ , by  
\cite[p.600]{ka}, the solution of (\ref{k.2.6}) is  
 
\begin{equation}\label{k.2.7} 
J(x)=\exp\left(-\int_{0}^{x}sAds\right)J(0)=\exp \left(-\frac{Ax^2}{2}\right)\frac 
{1}{2}\left(\frac{\pi A}{2}\right)^{\frac{1}{2}} \ \ . 
\end{equation} 
 
This proves the equality (\ref{k.2.2}). \hspace{0.5cm} $\Box$ 
\vspace{1mm} 
 
Under the hypothesis (\ref{k.2.1}), by the Leibnitz rule for the differentiation 
of parametric matrix integrals \cite[p.174]{01}, and \cite{20}, one gets 
 
\begin{equation}\label{k.2.8} 
\frac{d^{2n}}{dx^{2n}}J(x) = (-1)^{n}2^{2n-1}\int_{-\infty}^{+\infty} \exp 
\left(-2A^{-1}v^2\right)v^{2n} \, e^{2ixv}dv \ \ . 
\end{equation} 
 
Using Rodrigues's formula for the Hermite matrix polynomials \cite{15}, it 
follows that 
 
\begin{equation}\label{k.2.9} 
H_{2n}(x,A)=\exp\left(\frac{Ax^2}{2}\right)\left(\frac{A}{2}\right)^{-n}\left[ 
\frac{d^{2n}}{dx^{2n}}\left(\exp \left( -\frac{Ax^2}{2}\right)\right)\right] \ \ . 
\end{equation} 
 
By (\ref{k.2.2}) and (\ref{k.2.9}) and the fact 
 
$$ 
\int_{-\infty}^{+\infty} \exp \left(-2A^{-1} 
v^2\right)v^{2n}\sin{2vx}dv=0 \ \ , 
$$ 
 
it follows that 
 
\begin{eqnarray}\label{k.2.10} 
H_{2n}(x,A) & = & 2 \left(\frac{2A^{-1}}{\pi}\right)^{\frac{1}{2}} 
                  \exp\left(\frac{Ax^2}{2}\right)\left(\frac{A}{2}\right)^{-n} 
                  (-1)^n2^{2n-1}\int_{-\infty}^{+\infty} \exp \left(-2A^{-1} 
                  v^2\right)v^{2n}e^{2ivx}dv \nonumber \\ 
            & = & \frac{1}{\sqrt{\pi}}\exp\left(\frac{Ax^2}{2}\right) 
                  \left(\frac{A}{2}\right)^{-(n+\frac{1}{2})}(-1)^n2^{2n} 
                  \int_{-\infty}^{+\infty}\exp \left(-2A^{-1}v^2\right)v^{2n} 
                  e^{2ivx}dv \nonumber \\  
            & = & \frac{\exp\left(\frac{Ax^2}{2}\right)}{\sqrt{\pi}}\left(\frac{A}{2} 
\right)^{-(n+\frac{1}{2})}(-i)^{2n}2^{2n}\int_{-\infty}^{+\infty} \exp \left( 
-2v^2A^{-1}+2vixI \right)v^{2n}dv \ \ . 
\end{eqnarray} 
 
Similarly, for odd indices we have 
 
\begin{equation}\label{k.2.11} 
\frac{d^{2n+1}}{dx^{2n+1}}J(x) = (-1)^{n+1}\frac{2^{2n}}{i}\int_{-\infty}^{ 
+\infty}\exp\left(-2A^{-1}v^2\right)v^{2n+1} \, e^{2ixv}dv \ \ . 
\end{equation} 
 
By the Rodrigues formula for the Hermite matrix polynomials \cite{15}, 
 
\begin{equation}\label{k.2.12} 
H_{2n+1}(x,A)=-\exp\left(\frac{Ax^2}{2}\right)\left(\frac{A}{2} 
\right)^{-(n+\frac{1}{2})}\left[\frac{d^{2n+1}}{dx^{2n+1}}\left(\exp\left(- 
\frac{Ax^2}{2}\right)\right)\right] \ \ , 
\end{equation} 
 
so from (\ref{k.2.2}), (\ref{k.2.11}) and (\ref{k.2.12}) and the fact that  
$(-i)^{2n+1}=i(-1)^{n+1}$,  
 
\begin{equation}\label{k.2.13} 
H_{2n+1}(x,A) = \left(\frac{A}{2}\right)^{-(n+1)}\frac{(-2i)^{2n+1}} 
                {\sqrt{\pi}}\exp\left(\frac{Ax^2}{2}\right) 
                \int_{-\infty}^{+\infty}\exp\left(-2v^2A^{-1} 
                + 2ixvI \right)v^{2n+1}dv \ \ . 
\end{equation} 
 
Note that formulas (\ref{k.2.10}) and (\ref{k.2.13}) can be written as a single 
formula 
 
\begin{equation}\label{k.2.14} 
H_n(x,A)=\left(\frac{A}{2}\right)^{-\left(\frac{n+1}{2}\right)}\frac{2^n(-i)^n 
\exp\left(\frac{Ax^2}{2}\right)}{\sqrt{\pi}}\int_{-\infty}^{+\infty}\exp\left(-2v^2A^{-1} 
+2ivxI\right)v^ndv \ \ . 
\end{equation} 
 
Summarizing, the following integral representation for the Hermite matrix 
polynomials has been proved: 
 
\begin{theorem}\label{k.t.2.3} Let $A$ be a matrix in $C^{r \times r}$ 
satisfying condition (\ref{k.2.1}) and let $H_n(x,A)$ be the $nth$ 
Hermite matrix polynomial. Then the integral representation (\ref{k.2.14}) 
holds for all $n \geq 0$ and $-\infty < x < +\infty$. 
 
\end{theorem} 
 
\vspace{1mm} 
 
\underline{REMARK 1:} For the case $r=1$, and $A=2$, expression (\ref{k.2.14}) 
coincides with formula (\ref{k.1.1}). 
 
 
\section{On the exact computation of certain matrix integrals.} 
 
Let us consider the matrix integral 
 
\begin{equation}\label{k.4.1} 
{\cal Y}=\int_{-\infty}^{+\infty}\exp\left(Bx\right)\exp\left(-\frac{Ax^2} 
{2}\right)dx \ \ , 
\end{equation} 
 
where $A$ and $B$ are matrices in $C^{r \times r}$ and $A$ satisfies the 
condition (\ref{k.2.1}). To compute ${\cal Y}$, we will use the parametric 
matrix integral 
 
\begin{equation}\label{k.4.2} 
Y(t)=\int_{-\infty}^{+\infty}\exp\left(Bxt\right)\exp\left(-\frac{Ax^2}{2} 
\right)dx \ \ , \ \ -\infty < t < +\infty \ \ , 
\end{equation} 
 
which, under the condition (\ref{k.2.1}), by (\ref{k.1.4}), is pointwise  
convergent for any real number $t$. As well, the matrix integral 
 
$$ 
{\cal Y}_0=\int_{-\infty}^{+\infty}\frac{d}{dt}\left(\exp\left(Bxt\right)\exp\left(-\frac{Ax^2}{2} 
\right)\right)dx \ \ , 
$$ 
 
may be written in the form 
 
\begin{equation}\label{k.4.3} 
{\cal Y}_0=\int_{-\infty}^{+\infty}Bx\exp\left(Bxt\right)\exp\left(-\frac{Ax^2}{2} 
\right)dx \ \ . 
\end{equation} 
 
Furthermore, from the inequality (\ref{k.1.4}) applied to both  
$\exp\left(Bxt\right)$ and $\exp\left(-\frac{Ax^2}{2}\right)$, the parametric matrix  
integral (\ref{k.4.3}) 
is uniformly convergent in every neighbourhood of $t$ lying in the real line. 
Thus, by the Leibnitz theorem for differentiation of matrix 
integrals \cite[p.174]{01}, and taking into account that $A$ commutes with 
$\exp\left(-\frac{Ax^2}{2}\right)$, it follows that 
 
\begin{equation}\label{k.4.4} 
Y'(t)=-B\int_{-\infty}^{+\infty} \exp\left(Bxt)\right)\left\{-Ax\exp\left( 
            -\frac{Ax^2}{2}\right)\right\}dx A^{-1} \ \ . 
\end{equation} 
 
From application of integration by parts in (\ref{k.4.4}), one gets 
 
\begin{equation}\label{k.4.5} 
Y'(t) = -B \left\{ \left[ \exp \left( Bxt \right) \exp \left( 
            -\frac{Ax^2}{2} \right) \right]_{x=-\infty}^{+\infty}   
      - Bt \int_{-\infty}^{+\infty} \exp \left( Bxt \right) \exp 
            \left( -\frac{Ax^2}{2} \right) dx \right\} A^{-1} \ \ . 
\end{equation} 
 
Under the condition (\ref{k.2.1}), from (\ref{k.1.4}) it follows 
 
\begin{equation}\label{k.4.6} 
\lim_{x \rightarrow \pm \infty} \exp \left( Bxt \right) \exp \left( -\frac 
{Ax^2}{2} \right) =0 \ \ , \ \ -\infty < t < +\infty \ \ , 
\end{equation} 
 
and by (\ref{k.4.5}) and (\ref{k.4.6}), 
 
\begin{equation}\label{k.4.7} 
Y'(t)=B^2tY(t)A^{-1} \ \ , \ \ -\infty < t < +\infty \ \ . 
\end{equation} 
 
Note that taking $t=0$ in (\ref{k.4.2}) and by the expression (4.8) of 
\cite{15}, we have 
 
\begin{equation}\label{k.4.8} 
Y(0)=\int_{-\infty}^{+\infty} \exp\left(-\frac{Ax^2}{2}\right)dx=\left(2 \pi 
A^{-1}\right)^{\frac{1}{2}} \ \ . 
\end{equation} 
 
From the previous comments it is easy to establish the following result: 
 
\begin{theorem}\label{k.t.4.1} Let $A$ and $B$ be commuting matrices in 
$C^{r \times r}$ for which $A$ satisfies the condition (\ref{k.2.1}). Then, the 
matrix integral (\ref{k.4.2}) may be written 
 
\begin{displaymath} 
Y(t)=\int_{-\infty}^{+\infty} \exp\left(Bxt\right) \exp\left(-\frac{Ax^2}{2} 
\right)dx=\left(2 \pi A^{-1}\right)^{\frac{1}{2}} \exp \left(\frac{t^2B^2 
A^{-1}}{2}\right) \ \ . 
\end{displaymath} 
\end{theorem} 
\vspace{1mm} 
 
\section{On the generating function of products of Hermite matrix 
polynomials.} 
 
In this section we extend formula (\ref{k.1.2}) to the matrix framework, using 
the integral representation of the Hermite matrix polynomials obtained in 
section 2. According to (\ref{k.2.14}), for $|t|<1$ we have 
 
$$ 
\sum_{n \geq 0}\frac{H_n(x,A)H_n(y,A)}{2^n \, n!}t^n 
$$ 
 
\begin{eqnarray}\label{k.5.0} 
= & \left(\frac{A}{2}\right)^{-1}\frac{ \exp \left( \frac{A}{2}(x^2+y^2) 
           \right)}{\pi} \sum_{n \geq 0} \frac{ \left(-4tA^{-1}\right)^n}{n!} 
           \nonumber \\ 
\times & \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \exp \left( -2A^{ 
           -1}(u^2+v^2)\right) \exp \left(2iI(ux+vy)\right)(uv)^n dudv \ 
           \ . 
\end{eqnarray} 
 
Note that by the inequality (\ref{k.1.4}), for each positive integer $n$, 
the matrix function 
 
\begin{equation}\label{k.5.111} 
\Phi_n(u,v)=\sum_{h=0}^n\frac{\left(-4tuvA^{-1}\right)^h}{h!}\exp \left( 
-2A^{-1}(u^2+v^2)+2Ii(ux+vy)\right) 
\end{equation} 
 
is integrable in ${\cal R}^2$. If we 
denote by $\Phi(u,v)$ the non-negative function defined by 
 
\begin{equation}\label{k.5.112} 
\Phi(u,v)=\exp \left(4 |u||v|\|A^{-1}\|-2\alpha(A^{-1})(|u|^2+|v|^2) \right) 
\sum_{k=0}^{r-1}\frac{\left[2 
\sqrt{r}(|u|^2+|v|^2)\|A^{-1}\|\right]^k}{k!} \ \ , 
\end{equation} 
 
for $(u,v) \in {\cal R}^2$, then by 
(\ref{k.1.4}), (\ref{k.5.111}) and (\ref{k.5.112}), it follows that 
 
$$ 
\|\Phi_n(u,v)\| \leq \Phi(u,v) \ \ , \ \ (u,v) \in {\cal R}^2 \ \ , \ 
\ n \geq 0 \ \ . 
$$ 
 
From (\ref{k.2.5}), it is clear that $\alpha(A^{-1})>0$ and $\Phi(u,v)$ 
is integrable in ${\cal R}^2$.  
By the dominated convergence theorem \cite[p.83]{fo}, we can change  
the order of summation and integration in (\ref{k.5.0}), obtaining 
 
$$ 
\sum_{n \geq 0}\frac{H_n(x,A)H_n(y,A)}{2^n \, n!}t^n =\left(\frac{A}{2} 
\right)^{-1}\frac{ \exp \left( \frac{A}{2}(x^2+y^2) 
\right)}{\pi} 
$$ 
 
\begin{equation}\label{k.5.2} 
\times \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} 
\exp\left(-2A^{-1}(u^2+v^2)\right) \exp \left(2iI(ux+vy)\right) \exp \left( 
-4tuvA^{-1}\right) dudv \ \ . 
\end{equation} 
 
Using the Fubini-Tonelli theorem \cite[p.65]{f1} we can write 
 
$$ 
\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} 
\exp\left(-2A^{-1}(u^2+v^2)\right) \exp \left(2iI(ux+vy)\right) \exp \left( 
-4tuvA^{-1}\right) dudv 
$$ 
 
\begin{equation}\label{k.5.3} 
= \int_{-\infty}^{+\infty} \exp\left(-2A^{-1}v^2+2ivyI)\right)\left\{ 
   \int_{-\infty}^{+\infty} \exp \left(-2A^{-1}u^2+2u \left(ixI-2tA^{-1} 
   v\right)\right) du \right\} dv \ \ . 
\end{equation} 
 
Let us consider the matrix integral 
 
\begin{equation}\label{k.5.4} 
L=\int_{-\infty}^{+\infty} \exp \left(-2A^{-1}u^2+2u \left(ixI-2tA^{-1}v \right) 
\right) du \ \ ; 
\end{equation} 
 
after the change of variable $\omega=2u$, this can be written in the form 
 
\begin{equation}\label{k.5.5} 
L=L(1)=\frac{1}{2}\int_{-\infty}^{+\infty} \exp \left(B_{0}\omega\right) 
\exp \left(-\frac{\omega^2}{2}A^{-1}\right)d\omega \ \ , 
\end{equation} 
 
where 
 
\begin{equation}\label{k.5.6} 
B_{0}=ixI-2tA^{-1}v 
\end{equation} 
 
Since $B_{0}$ commutes with $A^{-1}$, and under the hypothesis (\ref{k.2.1}) 
its eigenvalues  satisfy (\ref{k.2.5}). By (\ref{k.5.5}) and (\ref{k.5.6}),  
where $A^{-1}$ plays the role of $A$, and $t$ is replaced by 
$1$, it follows that 
 
\begin{equation}\label{k.5.7} 
L=\left(\frac{\pi A}{2}\right)^{\frac{1}{2}}\exp \left(-\frac{Ax^2}{2}\right) 
\exp \left(2t^2v^2A^{-1}-2txviI\right) \ \ . 
\end{equation} 
 
By (\ref{k.5.7}) and (\ref{k.5.3}), 
 
$$ 
\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \exp \left(-2A^{-1}(u^2+ 
v^2)\right) \exp \left( 2ix(u+v)I \right) \exp \left(-4tuvA^{-1} \right)dudv 
$$ 
 
\begin{equation}\label{k.5.8} 
=\left(\frac{\pi A}{2}\right)^{\frac{1}{2}} \exp \left(\frac{-x^2A}{2} 
\right) \int_{-\infty}^{+\infty} \exp \left(-2v^2A^{-1}(1-t^2)+2vi(y-tx) 
\right) dv \ \ . 
\end{equation} 
 
Let us consider now the matrix integral 
 
\begin{equation}\label{k.5.9} 
S=\int_{-\infty}^{+\infty} \exp \left(-2v^2A^{-1}(1-t^2)+2vi(y-xt)I 
\right)dv \ \ . 
\end{equation} 
 
Under the change of variable $\omega=2v\sqrt{1-t^2}$ in (\ref{k.5.9}),  
 
\begin{eqnarray}\label{k.5.10} 
S&= & \frac{1}{2\sqrt{1-t^2}}\int_{-\infty}^{+\infty} \exp \left( 
            B_{1}\omega (y-xt) \right) \exp \left( -\frac{ 
            \omega^2A^{-1}}{2}\right)d\omega \ \ , 
\end{eqnarray} 
 
where 
 
\begin{equation}\label{k.5.11} 
B_{1}=\frac{iI}{\sqrt{1-t^2}} \ \ . 
\end{equation} 
 
By (\ref{k.5.10}) and (\ref{k.5.11}), where $A^{-1}$ 
plays the role of $A$, and $y-xt$ replaces $t$, it follows that 
 
\begin{equation}\label{k.5.12} 
S=\frac{\left(2 \pi A \right)^{\frac{1}{2}}}{2\sqrt{1-t^2}} \exp \left(\frac 
{(y-xt)^2B_{1}^2A}{2}\right)=\left(\frac{\pi A}{2(1-t^2)}\right)^{\frac 
{1}{2}} \exp \left( \frac{-A(y-xt)^2}{2(1-t^2)}\right) \ \ . 
\end{equation} 
 
Consequently, by (\ref{k.5.8}), (\ref{k.5.10}) and (\ref{k.5.12}),  
 
$$ 
\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \exp \left(-2A^{-1} 
(u^2+v^2)\right) \exp \left(2iI(ux+vy)\right) \exp \left(-4tuvA^{-1} 
\right) dudv 
$$ 
 
\begin{eqnarray}\label{k.5.13} 
=\frac{\pi A}{2\sqrt{1-t^2}} \exp \left(-\frac{Ax^2}{2}\right) \exp \left(- 
\frac{A(y^2+x^2t^2-2yxt)}{2(1-t^2)}\right) \ \ . 
\end{eqnarray} 
 
Hence, from (\ref{k.5.2}) and (\ref{k.5.13})  
 
$$ 
\sum_{n \geq 0}\frac{H_n(x,A)H_n(y,A)}{2^n \, n!}t^n =(1-t^2)^{-\frac{1}{2}}  
\exp \left(A\frac{\left[xyt-\frac{1}{2}(x^2+y^2)t^2 \right]}{(1-t^2)} \right) \ \ . 
$$ 
 
Thus, the following result has been established: 
 
\begin{theorem}\label{k.t.5.1} 
Let $A$ be a matrix in $C^{r \times r}$ satisfying the condition (\ref{k.2.1}), 
and let $H_n(x,A)$ be the $nth$ Hermite matrix polynomial. Then, for any real 
value of  $t$, with $|t|<1$, and for any real values of $x$ and $y$, 
 
\begin{eqnarray}\label{k.5.14} 
W(x,y,t,A) & = & (1-t^2)^{-\frac{1}{2}} \exp \left( \frac{A}{2}\frac{ 
                 \left[2xyt-(x^2+y^2)t^2\right]}{(1-t^2)}\right) \nonumber \\ 
           & = & \sum_{n \geq 0}\frac{H_n(x,A)H_n(y,A)}{2^n \, n!}t^n \ 
                 \ , \ \ |t|<1 \ \ . 
\end{eqnarray} 
 
\end{theorem} 
 
\vspace{1mm} 
 
\underline{REMARK 2:} For the case $r=1$, and taking $A=2$, the formula 
(\ref{k.5.14}) coincides with (\ref{k.1.2}). 
 
\vspace*{1cm} 
 
\underline{ACKNOWLEDGEMENTS:} This 
paper has been supported by the Spanish D.G.I.C.Y.T grant PB93-381 
and the {\em Generalitat Valenciana} grant GV-C-CN-10-057-96. 
 
  
\vspace*{0.3cm} 
 
 
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 {\sc 
\begin{flushleft} 
Lucas J\'odar and Emilio Defez \\ 
Departamento de Matem\'atica Aplicada.\\ 
Universidad Polit\'ecnica de Valencia.\\ 
P.O.BOX 22.012. Valencia, Spain.\\ 
{\sf E-mail:}{\tt  ljodar@upvnet.upv.es}\\ 
{\sf E-mail:}{\tt edefez@mat.upv.es }
\end{flushleft}  }
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