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\begin{document}
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\markboth{\sc \centerline{Proceedings of the IWOP'96} }{\sc \centerline{Proceedings of the IWOP'96} }


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\input referencia

\begin{center}
{ \large \bf ON FACTORIZATION AND ALGEBRAIZATION OF  DIFFERENCE EQUATIONS 
OF HYPERGEOMETRIC TYPE}
\end{center}
\begin{center}
{   \bf Yu. F. Smirnov  }
\end{center}

 \begin{abstract}
The ``connection'' between the approach of Nikiforov, Suslov and Uvarov 
and the factorization method of Infeld and Hull to find explicit solutions
of finite difference equations of  hypergeometric type is discussed. The
group theoretical interpretation of the dynamical $su(2)$ algebra for
Kravchuk functions is described.
\end{abstract}

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\section{Introduction}

In Ref. \cite{1}-\cite{1-1} two questions were arised concerning to the theory of
finite difference equations of the hypergeometric type   \cite{2}-\cite{2-2}.

\begin{description}

\item \cite{1}-\cite{1-1} Using the case of Kravchuk polynomials $k^p_n(s,N)$ as an
example it was shown that the corresponding finite difference
equation is equivalent to the eigenvalue problem

\be
H\psi_n = - \lambda_n \psi_n
\ee
where the ``hamiltonian' $H$ is factorized
%1.2
\be
H= -a^+a
\ee
and $a^+, a$ are constructed from standard first order difference
operators 
%1.3
\be
T_\pm = \exp (\pm \partial/\partial s).
\ee
This fact allows to apply to such type of equations the methods
elaborated for the solution of equations with factorized hamiltonians
 \cite{3} as it was suggested at first in the paper  \cite{4}.

\item  \cite{2}-\cite{2-2} Besides $a^+$ and $a$ two other operators $A^+, A$ were
constructed in  \cite{1}-\cite{1-1} which form a closed algebra together with the
Hamiltonian $H$. 

They satisfy the commutation relations 
%1.4
\bea
[H, A^+] &=& A^+, [H, A]=-A,\nonumber\\
\, [A^+, A] &=& H,
\eea
{\it i.e.}, they generate the SU(2) algebra. Thus in the space of
Kravchuk functions
%1.5
\be
\psi_n(s) = \sqrt{\rho(s)} k^p_n (s,N), \; n=0,1,\dots, N,
\ee
where $\rho(s)$ is the usual weight function, the irreducible
representation $D^j$ of the SU(2) algebra is realized whose dimension
$2j+1$ is equal to $N+1$. Therefore in  \cite{1}-\cite{1-1} the dynamical algebra
SU(2) for Kravchuk functions was found. It seems to be the first
example of dynamical algebras for finite difference equations.

\end{description}



Since Ref.  \cite{1}-\cite{1-1} concerns only Kravchuk polynomials (functions) it is
interesting to study the problem of factorization and algebraization
for other types of classical orthogonal polynomials of a discrete
variable. In this paper we want to demonstrate (in sections 2 and 3)
that the factorization found in  \cite{1}-\cite{1-1} takes place also for all
classical orthogonal polynomials of a discrete variable and a close
connection exists between the approach to the finite difference
equations of the  hypergeometric type, developed in Refs.  \cite{2}-\cite{2-2}, and the 
factorization method  \cite{3} for the solution of second order
differential equations. For simplicity we restrict ourselves in these
sections by the discussion of finite difference equations on the
uniform lattice. In the section 4 the group theoretical interpretation
of the dynamical SU(2) algebra for Kravchuk  functions is
suggested. 

It should be noted that in the general case the dynamical algebra does
not coincide with any usual Lie algebra, it can be of more general
type (quadratic or higher power algebra)  \cite{5}.

\section{Factorization of the finite difference equations of the
hypergeometric type}

\setcounter{equation}{0}

The finite difference equation of the hypergeometric type is of the
form   \cite{2}-\cite{2-2}
%2.1
\be
\bigg(\sigma(s) \nabla\Delta + \tau (s) \Delta + \lambda_n\bigg) y_n (s) =0.
\ee
Here $y_n(s)$ is of the polynomial of the power $n$ with respect to
the discrete variable $s$, $\sigma(s)$ and $\tau(s)$ are the polynomials
of the second and first degree, respectively, and
%2.2
\be
\Delta = T_+ - 1 \quad {\rm and}\quad \nabla = 1-T_-.
\ee
%
As it was mentioned above we consider here only the case of the
uniform lattice $s=0,1,\dots, N$, with a finite number of nodes --$N+1$--.
The results obtained below can be extended easily on the case of
infinite grid $s=0,1,2,\dots$ (Meixner and Charlier polynomials).

The Eq. (2.1) is equivalent to the recurrent relation.
%2.3
\be
A_s y_n (s+1) + B_s y_n (s) + C_s y_n (s-1) + \lambda_n y_n (s)=0
\ee
where
%2.4
\be
A_s= \sigma(s) + \tau (s), \quad C_s= \sigma(s), \quad B(s) = -
(A_s+C_s).
\ee
Therefore the solution of the Eq. (2.1) corresponds to the
diagonalization of the tridiagonal matrix $\calH^{(0)}$ with the
following nonvanishing matrix elements
%2.5
\be
\calH^{(0)}_{ss} = B_s, \quad \calH^{(0)}_{s+1,s} = A_s, \quad 
\calH^{(0)}_{s-1,s} = C_s, \quad  s=0,1,\dots, N.
\ee
This matrix is nonhermitian. In order to operate with the hermitian
``Hamiltonian'' $H^{(0)}$ instead of ${\cal H}^{(0)}$ it is
reasonable to go, following to  \cite{1}-\cite{1-1}, from polynomial $y^{(0)}_n (s) \equiv
y_n(s)$ to ``functions''
%2.6
\be
\psi^{(0)}_n (s) = S_0 y_n^{(0)} (s)
\ee
using the similarity transformation
%2.7
\be
S_0 = \sqrt{\rho (s)}.
\ee
where $\rho(s)$ is an usual weight function for polynomials $y_n(s)$.
Therefore the orthogonality relations for the functions (2.6) is of
the form 
%2.8
\be
\langle \psi_{n'} | \psi_n\rangle \equiv \sum^N_{s=0}
  \psi_n(s) \psi_{n'}(s)  = \delta_{nn'} d^2_n.
\ee
As a consequence, the matrix ${\cal H}^{(0)}$ transforms into the hermitian
tridiagonal matrix
%2.9
\be
H^{(0)} = S_0 \calH^{(0)} S^{-1}_0
\ee
with nonvanishing matrix elements
%2.10
\be
H^{(0)}_{ss} = B_s, \; H^{(0)}_{s+1s} = H^{(0)}_{ss+1} = \sqrt{A_sC_{s+1} }.
\ee
The recurrent relation (2.3)  takes the form
%2.11
\be
\sqrt{A_s C_{s+1}} \psi^{(0)}_n (s+1) + B_s 
\psi^{(0)}_n (s) + \sqrt{A_{s-1} C_s} 
\psi^{(0)}_n (s-1) + \lambda^{(0)}_n \psi^{(0)}_n (s)=0.
\ee
We can rewrite equation (2.12) as the eigenvalue problem
%2.12
\be
H^{(0)} \psi^{(0)}_n (s)= - \lambda^{(0)}_n \psi^{(0)}_n (s),
\ee   
for the Hamiltonian
%2.13
\be
H^{(0)} = \sqrt{A_s C_{s+1} } T_+ + B_s + T_- \sqrt{A_s C_{s+1} }.
\ee
The above Hamiltonian (2.13) is hermitian if the following conjugation rule 
%2.14
\be
(T_-)^\dagger= T_+
\ee
is accepted for the finite difference operators $T_+$ and $T_-$.

Let us turn now to the factorization of the Hamiltonian $H^{(0)}$. As
it was mentioned in   \cite{2}-\cite{2-2} the initial operator $\calH^{(0)}$ can  be easily
factorized  
%2.15
\be
\calH^{(0)} = -b^+ b,
\ee
where
%2.16
\bea
b=\Delta, & \hspace{1cm} & b^+=- (\sigma (s) \nabla + \tau (s)) = C_s T_- - A_s.
\eea
To factorize the Hamiltonian $H^{(0)}$ we apply to the expression
(2.15) the similarity transformation
%2.17
\be
H^{(0)} = -S_0 b^+ S^{-1}_1 S_1 b S^{-1}_0
\ee
with
%2.18
\be
S_1 = \sqrt{\rho_1 (s) }.
\ee
Then, we obtain
%2.19
\be
H^{(0)} = -a^+ a,
\ee
where
%2.20
\bea
a &=& S_1 b S^{-1}_0 = \sqrt{\frac{\rho_1(s)}{\rho(s+1)} } T_+ -
\sqrt{\frac{\rho_1(s)}{\rho(s)} }, \nonumber \\
a^+ &=& S_0 b^+ S^{-1}_1 = \sqrt{\frac{\rho(s)}{\rho_1(s-1)} } C_s
T_- - \sqrt{\frac{\rho(s)}{\rho_1(s)} } A_s . 
\eea
Taking account the relation   \cite{2}-\cite{2-2}
%2.21
\bea
\frac{\rho(s+1)}{\rho(s)} &=& \frac{A_s}{C_{s+1} }, \\
\rho_1(s)  &=& \rho (s+1) C_{s+1}
\eea
we can rewrite the formulas (2.20) as follows
%2.23
\bea
a = \sqrt{C_{s+1}} T_+ - \sqrt{ A_s}, & \hspace{1cm} & a^+ =\sqrt{C_s} T_- - \sqrt{ A_s} .
\eea
These operators are an hermitian conjugate pair in the sense of Eq.
(2.14). The results (2.19) and (2.23) are equivalent to ones from  \cite{1}-\cite{1-1}
in the case of Kravchuk functions. Thus the factorization mentioned
in  \cite{1}-\cite{1-1} is valid for all finite difference equations of the
hypergeometric type on the uniform grid.

Now we shall demonstrate that the method developed in   \cite{2}-\cite{2-2}
to solve
such type of equations has a deep parallelism with the factorization
method  \cite{3} if we work with the functions (2.6) instead of polynomials
$y_n(s)$. 

\section{Factorization method and classical orthogonal polynomials of
a discrete variable}

\setcounter{equation}{0}

To begin with we remind the main points of the factorization method
(FM)  \cite{3} adopted to our problem.

In the FM has been considered  a finite chain of hermitian Hamiltonians
%3.1
\be
H^{(0)},  H^{(1)},  \dots , H^{(m)},  \dots , H^{(l)}.
\ee
Each of these Hamiltonians is factorized as follows
%3.2
\be
H^{(m)} = - a^+ (m, m+1) a (m+1, m).
\ee
Let the initial Hamiltonian $H^{(0)}$ has a number  (finite or
infinite) of discrete levels with eigenfunctions $\psi^{(0)}_n$ 
%3.3
\be
H^{(0)} \psi^{(0)}_n = \lambda^{(0)}_n \psi^{(0)}_n .
\ee
We select one of them with the function $\psi^{(0)}_l (l=0,1,2,\dots)$ and
form the chain of functions
%3.4
\be
\psi^{(0)}_l,  \psi^{(1)}_{l-1},  \dots , \psi^{(l-m)},  \dots , \psi^{(l)}_0 
\ee
where $\psi^{(m)}_k$ is an eigenfunction of the Hamiltonian
%3.5
\be
H^{(m)} \psi^{(m)}_k =  \lambda^{(m)}_k \psi^{(m)}_k .
\ee
The neighbouring functions in the chain (3.4) are related to each other 
by the following expressions involving the operators $a^+,a$  (3.2) 
%3.6
\be\begin{array}{rcl}
a(m+1, m) \psi^{(m)}_{l-m} &=& \sqrt{\lambda^{(m)}_{l-m} }
\psi^{(m+1)}_{l-m-1} , \\
a^+(m, m+1) \psi^{(m+1)}_{l-m-1} &=& \sqrt{\lambda^{(m)}_{l-m} }
\psi^{(m)}_{l-m} .\end{array}
\ee

The last function in (3.4)  has the sense of a ``vacuum" vector because
it is annihilated by the lowering operator $a$ 
%3.7
\be
a (l+1, l) \psi^{(l)}_{0} =0 .
\ee
It is clear from (3.6)-(3.7) that any function from the chain (3.4)
can be obtained from the vacuum vector using the raising operator
%3.8
\be
\psi^{(m)}_{l-m} = \frac{1}{\sqrt{\lambda^{(l)}_{0}
\lambda^{(l-1)}_{1}  \dots \lambda^{(m)}_{l-m} }} 
a^+(m,m+1) \dots  a^+ (l-1,l) \psi^{(l)}_0.
\ee
This function will be normalized to $1$ in the sense that 
%3.9
\be
\langle \psi^{m}_{l-m}|\psi^{(m)}_{l-m}\rangle = 1,
\ee
if the vacuum vector is correctly normalized, i.e.,
%3.10
\be
\langle \psi^{l}_{0}|\psi^{(l)}_{0}\rangle = 1.
\ee
The authors  \cite{3} have remarked that the main advantage of the FM is the fact
that it allows to calculate in a systematic way the eigenvalues and
the normalized eigenfunctions of various physical systems whose
Hamiltonian can be factorized as described above. In the 
original formulation the operators $a^+,a$ were first order differential operators
as well as  the  Hamiltonians $H^{(m)}$ were a second order differential operator
Now we apply this approach to the first order difference operators (2.16) and 
(2.23) and to the Hamiltonians given in (2.15) or (3.2). 
Notice that the algorithm developed in   \cite{2}-\cite{2-2} allows us to find  the solution 
$y^{(0)}_n(s)$ of the difference equation (2.1) and also their finite difference
derivatives defined by
%3.11
\be
y^{m}_{n}(s) = \Delta^m  y^{(0}_{n}(s) ,
\ee
which   satisfy the  following second order difference equation 
%3.13
\be
\calH^{(m)} y^{(m)}_k  \equiv \left[ \sigma(s) \Delta\nabla +\tau_m (s)
\Delta \right]  y^{(m)}_k (s)= - \lambda^{(m)}_k y^{(m)}_k (s),
\ee
where
%3.14
\bea
\tau_m (s) &=& \tau (s+m) + \sigma (s+m) - \sigma(s),\\
\lambda^{(m)}_k &=& \lambda^{(0)}_k + m\tau' + \frac{m(m-1)}{2}
\sigma'' . 
\eea
The constants $\tau'$ and $\sigma''$ can be found from the expressions
%3.15
\bea
\tau (s) = \tau' s +\tau (0), & \hspace{1cm} & \sigma (s)
= \ahalf \sigma'' s^2 +\sigma' (0) s+\sigma(0).
\eea
The polynomials $y^{(m)}_k(s)$ are defined at $s=0,1,2,\dots,N-m$.
They are orthogonal with the weight function
%3.16
\be
\rho_m(s) = \rho (s+m) \prod^m_{t=1} \sigma (s+t).
\ee

The operators $b, b^+$ play the role of lowering and raising operators
for these polynomials. In fact
%3.17
\bea
\Delta y^{(m)}_k &=& y_{k-1}^{(m+1)}\\
\bigg(\sigma(s) \nabla &+& \tau_m (s)\bigg) y_{k-1}^{(m+1)} = 
\lambda_{k}^{m} y_{k}^{(m)} .
\eea
Now, we can introduce, instead of polynomials (3.12), the corresponding
functions
%3.19
\bea
\psi^{(m)}_k = S_m y^{(m)}_k, \hspace{1cm} 
S_m = \sqrt{\rho_m (s) }.
\eea

They are normalized by the relation
%3.20
\be
\langle \psi^{(m)}_{k}|\psi^{(m)}_{k'}\rangle = \delta_{kk'} (d^m_k)^2.
\ee
where
%3.21
\be
(d^m_k)^2 = (d^{m+1}_k )^2/\lambda^{(m)}_{k} .
\ee

Similarly to Eqs. (2.17)-(2.23) we can introduce the hermitian
Hamiltonians
%3.22
\be
H^{(m)} = S_m \calH^{(m)} S^{-1}_m = \sqrt{A^m_s C^m_{s+1} } T_+ +
B^m_s + T_- \sqrt{A^m_s C^m_{s+1} } .
\ee
where
\[
 A^m_s =  \sigma (s) + \tau(s) ,\hspace{1cm}  C^m_s = C_s = \sigma (s) 
 \]
They can be factorized in the form
%3.23
\be
H^{(m)} = - a^+ (m,m+1) a(m+1,m)
\ee
with
%3.24
\bea
a^+(m,m+1) &=& \sqrt{C^m_s} T_- -\sqrt{A^m_s},\nonumber\\
a(m+1,m) &=& T_+ \sqrt{C^m_s}  -\sqrt{A^m_s},
\eea


In accordance with (3.17)-(3.19) we obtain
%3.24
\bea
a(m+1,m)  \psi^{(m)}_{k}  &=& \sqrt{  \lambda^{(m)}_{k} }
 \psi^{(m+1)}_{k-1} ,\nonumber\\
a^+(m,m+1) \psi^{(m)}_{k-1} &=& \sqrt{  \lambda^{(m)}_{k} }
 \psi^{(m)}_{k}.
\eea


Therefore,
%3.25
\be
H^{(m)} \psi^{(m)}_{k} = - a^+ (m,m+1) a (m+1,m) \psi^{(m)}_{k} =
 -\lambda^{(m)}_{k} \psi^{(m)}_{k},
\ee
as it should be in agreement with (3.12).

Now it is possible to compare the approach given   \cite{2}-\cite{2-2} with the 
FM  \cite{3}. For this aim we identify the finite difference operators
(3.23) with the Hamiltonian (3.2) from the FM, the function (2.6) 
 with the initial function $\psi^{(0)}_l$ in the chain (3.4) and
the function (3.19) at $k=l-m$ with the rest elements  of (3.4).
Notice that the operators $a_{\pm}$  having the properties (3.25),
play the same role as the lowering and raising operators (3.6)  in 
the FM. Thus we can consider the approach to finite difference equations of the
hypergeometric type developed in   \cite{2}-\cite{2-2} as an equivalent of the FM,
realized in terms of the finite difference operators. The explicit
expression for the function $\psi^{(0)}_l$ can be found using the Eq.
(3.7) of the FM approach which is an equivalent of the Rodrigues formula
for the solution of (2.1) or (3.12)   \cite{2}-\cite{2-2}. The normalization 
factor for $\psi^{(0)}_l$
 can be also calculated because all the necessary eigenvalues (3.14) are known.
Therefore, all is reduced to the calculation of the normalization
factor for the vacuum function
%3.26
\be
(d_0^l)^2 = \sum^{N-l}_{s=0} \rho_n(s),
\ee
which should be evaluated directly.

\section{Dynamical algebra for Kravchuk functions}

\setcounter{equation}{0}

In this section we want to give the group theoretical interpretation
of the dynamical SU(2) algebra for the Kravchuk functions, found in
Ref.  \cite{1}-\cite{1-1}. It is well known [3,8] that normalized Kravchuk function
%4.1
\be
\psi^p_n (s,N) = (d_n)^{-1} \sqrt{\rho (s)} k^p_n (s,N)
\ee
is proportional to the Wigner d-function
%4.3
\be
\psi^p_n (s,N) = (-1)^{m-m'}  d^j_{mm'}(\beta).
\ee
Here $j=N/2, n=j-m, s=j-m', p = \sin^2 \beta/2,
q=1-p=\cos^2\beta/2, d^2_n = N! (pq)^n/n! (N-n)!$ The difference
equation for d-functions can be obtained when we are seeking the
eigenfunctions $\phi_{jm}$ of the operator
%4.4
\be
J_{z'} =  \cos \beta J_z+ \sin \beta J_x
\ee
in terms of functions $|jm'>$, quantized along the axis $z$ forming
the angle $\beta$ with the axis $z'$.

Thus we solve the problem
%4.5
\be
J_{z'} \phi_{jm} = m \phi_{jm}
\ee
where
%4.6
\be
\phi_{m} = \sum_{m'} t_{m'} |jm'\rangle.
\ee
Substituting (4.5) into (4.4) and taking the scalar product of
the resulting expression by $<jm'|$ we obtain the recurrent relation
for coefficients $t_{m'}$
%4.7
\bea
\ahalf \sin \beta \langle jm'|J_+|jm'-1\rangle t_{m'-1} 
+\ahalf \sin \beta \langle jm'|J_-|jm'+1\rangle t_{m'+1} 
+  \cos  \beta m' t_{m'} = m t_{m'}.
\eea

It is well known [3,8] that coefficients $t_{m'}$ in (4.5) are
identical with $d^j_{mm'}(\beta)$. Thus (4.6) gives the finite
difference equation for d-functions or, taking into account the
relation (4.2), one for Kravchuk functions. As usually, the
operators $J_\pm = J_x \pm iJ_y$. Their matrix elements, appearing in
(4.6), are well known
%4.8
\be
\langle jm\pm 1|J_\pm|jm\rangle = \sqrt{(j\mp m) (j\pm m +1)}.
\ee
We are interesting in the dynamical algebra for the eigenvalue problem
(4.4). Therefore, we need the operators which transform the solution
$\phi_{jm}$ into $\phi_{jm\pm 1}$. Obviously the operators
%4.9
\be
J'_\pm = J_{x'} \pm i J_{y'}= \sin \beta J_z +\ahalf (\cos\beta \pm
1) J_+ + \ahalf (\cos\beta \mp 1) J_-,
\ee
which are, together with $J_{z'}$, the generators of the SU(2) algebra, 
can be taken as such operators because
%4.10
\be
J'_\pm \phi_{jm} = \sqrt{(j \mp m)(j\pm m+1)} \phi_{jm\pm 1}.
\ee
Thus the dynamical algebra for the eigenvalue problem (4.4) is found.
In order to transform it into the dynamical algebra for d-functions
or Kravchuk functions we should multiply the equation (4.9) by the
vector $<jm'|$. In such a manner we obtain
%4.11
\be\begin{array}{l}
  \sqrt{(j\mp m)(j\pm m+1)} d^j_{m\pm 1 m'} (\beta) 
= m' \sin \beta d^j_{mm'} (\beta) + \\
+ \ahalf (\cos \beta \pm 1) \langle jm'|J_+|jm'-1\rangle d^j_{mm'-1} (\beta)+ 
\ahalf (\cos\beta \mp 1) \langle jm'|J_-| jm'+1\rangle d^j_{mm'+1} (\beta).
\end{array}
\ee
The corresponding relations for the Kravchuk functions can be
written in the form
%4.12
\bea
A_+ \psi^p_n (s,N) &=& \sqrt{ (n+1)(N-n)} \psi^p_{n+1} (s,N),\\
A_- \psi^p_{n} (s,N) &=& \sqrt{ n(N-n+1)} \psi^p_{n-1} (s,N),
\eea
where
%4.15
\bea
A_+ &=& 2\sqrt{p q} (\frac{N}{2} - s) - p \sqrt{(s+1)(N-s)}
e^{-\partial_s} + q\sqrt{ s(N-s+1)} e^{\partial _s},\\
A_- &=& 2\sqrt{p q} (\frac{N}{2} - s) + q e^{-\partial_s} \sqrt{s(N-s+1)}
-pe^{\partial_s} \sqrt{ (s+1)(N-s)} .
\eea
These operators together with
%4.16
\be
A_0 = (q-p) (\frac{N}{2} -s) - \sqrt{pq} e^{\partial_s}
\sqrt{(N-s)(s+1)} - \sqrt{pq} e^{-\partial_s} \sqrt{s(N-s)},
\ee
are the generators of the dynamical SU(2) algebra found in  \cite{1}-\cite{1-1}. It is
easy to verify that they satisfy the commutation relations
%4.17
\be
[A_0, A_\pm] = \pm A_\pm, \; [A_+, A_-] = 2A_0,
\ee
which is a consequence  from the fact that the  operators $A_{\pm}$ and $A_0$, represent the
matrix realization of SU(2) generators $-J_{z'}, J_-'$ and $J'_+$
considered above. The Kravchuk functions (4.1) belong to the
irreducible representation $D^j, j=N/2$ of this algebra. 

\section{Conclusion}

It was shown that the approach to the solution of the finite
difference equations of  hypergeometric type developed by
Nikiforov, Suslov and Uvarov   \cite{2}-\cite{2-2}, becomes equivalent to the
factorization method of Infeld and Hull  \cite{3} after transition from the
classical orthogonal polynomials to the corresponding functions
introduced in  \cite{1}-\cite{1-1}.

The factorization of finite difference equations found in  \cite{1}-\cite{1-1} for the
Kravchuk functions can be extended on all classical orthogonal
polynomials of a discrete variable. The group theoretical
interpretation of the SU(2) dynamical algebra, constructed in  \cite{1}-\cite{1-1} for
Kravchuk functions, is suggested.

\section*{Acknowledgements}

The author is very thankful to Prof. N. M. Atakishiev for valuable
discussion and comments. This work was supported partially by the
Russian Foundation of the Fundamental Research, Grant No. 96-01-01421 

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\end{thebibliography}
{\sc
\begin{flushleft}
{ Yu. F. Smirnov ${}^{\dagger}$ } \\
{ Instituto de Ciencias Nucleares, UNAM.\\
Circuito Exterior C.U. A. Postal 70-543 04510  M\'exico, D. F. M\'exico } \\
{ {\sf E-mail:} {\tt smirnov@xochitl.nuclecu.unam.mx }}\\
{${}^{\dagger}$ On leave of abscence from Nuclear
Physics Institute, Moscow State University.}\\
\end{flushleft}
}
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