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\begin{center}
 {  \bf A CLASS OF JACOBI POLYNOMIALS ORTHOGONAL WITH RESPECT TO VARYING WEIGHTS.}
 \end{center}
\begin{center}
{\bf M.X. He, \quad P. Natalini, \quad and \quad P.E. Ricci }
\end{center}

\vskip .5cm 

\begin{abstract}   We introduce a wide class
of Jacobi Polynomials orthogonal with respect to varying weights, which are
connected with the Relativistic Hermite polynomials introduced by {\sc V.
Aldaya - J. Bisquert - J. Navarro-Salas} [1] for representing the wave functions
of the quantum relativistic harmonic oscillator.
\end{abstract}   

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\section{Introduction}

In recent papers the Relativistic Hermite Polynomials (RHP) have been
introduced by {\sc V. Aldaya - J. Bisquert - J. Navarro-Salas} [1], and studied
by {\sc J.S. Dehesa, A. Zarzo, A. Martinez, B. Nagel} 
[18], [19], [20], [21], [8].

These polynomials arise in the theory of the quantum relativistic harmonic
oscillator, for representing the wave functions of multidimensional
Schr\"odinger operators.

Some new sets of orthogonal polynomials, related to the RHP, and
consequently called Relativistic polynomials, have been introduced and
studied by the present authors, and by {\sc S. Noschese, K. Pan}.

These  are: the Relativistic Laguerre Polynomials (RLP), the Relativistic
Szeg\"o Polynomials (RSP), the Relativistic Jacobi Polynomials (RJP), the
Relativistic Bessel Polynomials (RBP).

Generally speaking all these polynomials satisfy the following properties:
\begin{enumerate}
\item[\footnotesize\#] they are connected with the RHP;
\item[\footnotesize\#] they depend on a parameter $N$ (which is related
with the speed of light, in case of the RHP) and approach to the classical  sets
of orthogonal polynomials when $N$ tends to infinity;
\item[\footnotesize\#] they are solutions of hypergeometric type equations;
\item[\footnotesize\#] they are orthogonal with respect to 
varying weights in suitable supports;
\item[\footnotesize\#] the varying weights satisfy a Pearson type differential
equation connecting all of them.
\end{enumerate}

\noindent It is worth to note that the RJP, RLP, RBP have been called
``relativistic'' only in accordance with the first of the above mentioned
properties, since no connection with  the wave functions of some quantum
relativistic harmonic oscillator has been proved until now.

We have studied
\begin{itemize}
\item Algebraic and differential properties, integral representations,
generating functions, orthogonality [4], [9], [10], [5], [13], [16], [3], [2].
\item Distribution of zeros (numerical approximation of 
the moments of the density of zeros) [17], [12], [14].
\item Asymptotic distribution of zeros [11], [6], [7].
\end{itemize}

\section{Recalling the Relativistic Hermite Polynomials}

In the above mentioned paper {\sc V. Aldaya - J. Bisquert - 
J. Navarro-Salas} [1] introduced a set of 
orthogonal polynomials in order to represent the wave function 
of the quantum relativistic harmonic oscillator, i.e.
$$
\tilde\psi=\tilde\psi_n(t,x)=e^{-in\omega t}\phi^N_n(x)=e^{-in\omega
t}\alpha^{-(N+n)}H^N_n(\xi(x))\ ,
$$
where $N:={mc^2\over \hbar\omega}$, $c$ denotes the speed of light,
$\hbar$ the Planck constant, $m$, $\omega$, denote physical parameters
(mass and frequency),
$$
\alpha:=\alpha(x):=\sqrt{1+{\omega^2\over c^2}\ x^2}\ ,
\hspace{1cm}
\xi:=\xi(x):=\sqrt{m\omega\over\hbar}\ x={\omega\over c}\sqrt{N}\ x\ ,
$$
$$
\phi^N_n(x)=\alpha^{-(N+n)}H^N_n(\xi(x))\ ,\qquad(\forall\,n\in 
\N_0 :=\N\cup\{0\})\ ,
$$
$H^N_n(\xi(x))$ are called Relativistic Hermite Polynomials, (RHP) and
constitute a family of polynomials satisfying the orthogonality property
(with respect to a varying weight function):
$$
\displaylines{
(\phi^N_n,\phi^N_m)=\int^{+\infty}_{-\infty}\alpha^{-(2N+n+m)}H^N_n
\,H^N_m\d x=2^n n!{(2N+n-1)!\over (2N-1)!}\ {1\over (2N)^n}\ {(2N-3)!!\over
(2N-2)!!}\sqrt{N}\ \pi\,\delta_{n,m}\ ,
\cr}
$$
($\delta = $ Kronecker delta).

Further properties of the RHP have been studied in [18] - [19] - [20] -
[8].\vskip1truecm

\section{New sets of  Polynomials orthogonal with respect to
varying weights}
\setcounter{equation}{0}

We have studied in the papers mentioned in section 1
some new sets of Polynomials orthogonal with respect to varying
weights and connected with the RHP. The main properties of these
polynomials,  which will be denoted here by the symbol
$G^{(N)}_n(x)$, can be easily summarized as follows.

They satisfy a differential equation (of hypergeometric type):
\begin{equation}\label{3.1}
\sigma(x)y''_n+\tau(n,x)y'_n+\lambda_ny_n=0\ ,
\end{equation}
where $\sigma$ is a polynomial of degree $\le 2$, $\tau$ is a
polynomial of degree $\le 1$, $\lambda_n$ is a constant, depending
on $n$, satisfying
$$
\lambda_n=-n\tau'-{n(n-1)\over 2}\sigma''
$$
(see the classical book of {\sc A.F. Nikiforov - V.B. Uvarov} [15]).

By changing $\sigma(x)$ in $z(1-z)$, a canonical hypergeometric
form can be obtained for each $G^{(N)}_n(x)$, (in particular the
{\sc B. Nagel} representation formula for the RHP can be found
in [8]):
$$
G^{(N)}_n(x)= k_n \, {}_2F_1(a_n,b_{n,N},c_{n,N};z)\ ,\quad (z=g(x))\ ,
$$
with a suitable choice of the constants $a_n$, $b_{n,N}$, $c_{n,N}$,
depending on the class of polynomials under consideration.

The symmetrization factor $\mu^{(N)}_n(x)$ by means of which
eq. (\ref{3.1}) can be rewritten in the self-adjoint form:
\begin{equation}\label{3.2}
\left(\mu_n^{(N)}(x)\sigma(x)y'_n\right)'-\mu_n^{(N)}(x)\lambda_ny_n=0\
,
\end{equation}
is the (varying) weight with respect to which the polynomials are
orthogonal.

The following Pearson-type equation connecting the weights holds
true:
\begin{equation}\label{3.3}
\left(\mu^{(N)}_n(x)\sigma(x)\right)'=\tau(n,x)\mu_n^{(N)}(x)\
.
\end{equation}
A Rodrigues formula can always be obtained
\begin{equation}\label{3.4}
G_n^{(N)}(x)={C_n\over \mu^{(N)}_n(x)}\ {\d^n\over \d
x^n}\left[\psi(x)(\mu^{(N)}_n(x))^n\right]\ ,
\end{equation}
where $C_n$ is a normalizing constant.

The orthogonality property reads
\begin{equation}\label{3.5}
\int_IG^{(N)}_n(x) G^{(N)}_m(x)\mu^{(N)}_{n,m}(x)\d
x=k_n\delta_{n,m}\ ,
\end{equation}
where $I$ is a suitable support, and the varying weight
$\mu^{(N)}_{n,m}(x)$ can be written
\begin{equation}\label{3.6}
\mu^{(N)}_{n,m}(x)=\varphi_n(x)\varphi_m(x)\mu^{(N)}(x)\
,
\end{equation}
with a suitable choice of functions $\varphi_n$ $(n\in \N_0)$, so that
the so called complete orthogonality becomes:
\begin{equation}\label{3.7}
\int_I\left[\varphi_n(x)G_n^{(N)}(x)\right]\
\left[\varphi_m(x)G^{(N)}_m(x)\right]\mu^{(N)}(x)\d
x=k_n\delta_{n,m} \ .
\end{equation}
A three-term recurrence relation can always be written in the
canonical form:
\begin{equation}\label{3.8}
A_{n,N}G^{(N)}_{n+1}(x)=\left[B_{n,N}-x\right]G_n^{(N)}(x)+
C_{n,N}G_{n-1}^{(N)}(x)\ , 
\end{equation}
where
$$
 G_{-1}^{(N)}(x):=0\ ;\ G_0^{(N)}(x):=1 \ .
$$

\section{A class of Jacobi Polynomials orthogonal with respect to varying
weights}

\setcounter{equation}{0}

In [5] a set of RJP $\{P_n^{(\alpha,\beta;N)}(x)\}_{n\in\N_0}$ has
been considered, by assuming the following definition properties:
\be\label{4.1}
\lim_{N\to\infty}P_n^{(\alpha,\beta;N)}(x)=P_n^{(\alpha,\beta)}(x)\
,\qquad(\forall\,n\in\N , \ \forall\,\beta > -1)
\ee
\be\label{4.2}
\lim_{\beta\to\infty}P_n^{(\alpha,\beta;N)}\left({1-{2x\over
\beta}}\right)=L_n^{(\alpha,N)}(x)\ ,\qquad(\forall\,n\in\N , \ 
\forall\, N > 1/2)
\ee
where $L_n^{(\alpha,N)}$ denote the corresponding RLP, defined in [9].

Consider the expression of the Jacobi polynomials
\be\label{4.3}
P_n^{(\alpha,\beta)}(x)=\sum^n_{k=0}{1\over k!}{n+\alpha\choose n-k}
(\alpha+\beta+n+1)_k\left({x-1\over 2}\right)^k\ ,
\ee
which can be easily obtained by the classical expression:
$$
P_n^{(\alpha,\beta)}(x)={1\over
n!}\sum^n_{k=0}(\alpha+k+1)_{n-k}(\alpha+\beta+n+1)_k{n\choose
k}\left({x-1\over 2}\right)^k\ ,
$$
taking into account
$$
{1\over k!}{n+\alpha\choose n-k}={1\over n!}(\alpha+k+1)_{n-k}{n\choose
k}\ .
$$
By putting ${1-2x\over \beta}$ instead of $x$ in formula (\ref{4.3}), and
letting $\beta\to\infty$, the classical limit property connecting the Jacobi to
the Laguerre polynomials follows:
$$
\lim_{\beta\to\infty}P_n^{(\alpha,\beta)}\left(1-{2x\over\beta}\right)=
L_n^{(\alpha)}(x)\
,\qquad (\forall\,n\in\N)\ .
$$
Conditions (\ref{4.1})-(\ref{4.2}) are satisfied by considering the explicit
expressions of the RLP (see [9])
\be\label{4.4}
L_n^{(\alpha,N)}(x)=\sum^n_{k=0}{n+\alpha\choose
n-k}{(-1)^k\left(-n-N+{1\over 2}\right)_k\over N^k}\ {(-x)^k\over k!}\ ,
\ee
and putting, by definition:
\be\label{4.5}
P_n^{(\alpha,\beta;N)}(x)=\sum^n_{k=0}{n+\alpha\choose
n-k}\left({N-\beta\over N}\right)^k\ \left[{AN-B\beta\over
N-\beta}\right]_k{1\over k!} \left({x-1\over 2}\right)^k\ ,
\ee
where the parameters $A,B$ must be chosen in such a way that conditions
(\ref{4.1}) and (\ref{4.2}) are satisfied.

This can be done (see [9]) by imposing two limit conditions
\be\label{4.6}
\left[{AN-B\beta\over
N-\beta}\right]_k\stackrel{(N\to\infty)}{\longrightarrow}
\left[\alpha+\beta+n+1\right]_k \ ,
\ee
\be\label{4.7}
\left[{AN-B\beta\over
N-\beta}\right]_k\stackrel{(\beta\to\infty)}{\longrightarrow}
\left[-n-N+{1\over 2}\right]_k \ .
\ee

To this aim, we put
$$A=\alpha+r\beta+n+1\ ;\qquad B=-n+{1\over 2}+sN\ ,$$
$$r-s=1+o(1)\quad (\hbox{\rm for }\{N\to\infty\} \ \hbox{ \rm  or } \
\{\beta\to\infty\})$$
$$(\hbox{\rm e.g.} \ o(1)=\rho_n(N,\alpha,\beta)=\rho(N,\beta)=
{p(n,\alpha)\over P(N,\beta)}\ ,$$
$$p,P\ \hbox{\rm polynomials and }\deg_N P(N,\beta)\ge 1 \ \hbox{\rm  and } 
\deg_\beta P(N,\beta)\ge 1) \ .$$

Then it is easily shown that conditions (\ref{4.6})-(\ref{4.7}) are satisfied.

Consequently we can introduce a general set of RJP generalizing  the set
introduced by {\sc M. He - P. Natalini}, which corresponds to the simplest 
choice: $\rho(N,\beta)\equiv 0$, by letting:
\begin{eqnarray}\label{4.8}
&&P_n^{(\alpha,\beta;N)}(\rho(N,\beta);x):=\sum^n_{k=0}{n+\alpha\choose
n-k}\left({n-\beta\over N}\right)^k{1\over
k!}\times\\
&&\qquad\times\left[{[\alpha(1+\rho(N,\beta))\beta+n+1]N+\left(n-{1\over
2}\right)\beta\over N-\beta}\right]_k\left({x-1\over 2}\right)^k\ .\nonumber
\end{eqnarray}
Then, the limit conditions (\ref{4.1})-(\ref{4.2}) hold true, since
\be\label{4.9}
\lim_{N\to\infty}P_n^{(\alpha,\beta;N)}(\rho(N,\beta);x)=
P_n^{(\alpha,\beta)}(x)\
,\qquad(\forall\,n\in\N , \ \forall\,\beta > -1)
\ee
\be\label{4.10}
\lim_{\beta\to\infty}P_n^{(\alpha,\beta;N)}\left(\rho(N,\beta);1-{2x\over
\beta}\right)=L_n^{(\alpha,N)}(x)\ ,\; (\forall\,n\in\N ,\ \forall\, N > 1/2)
\ee
By recalling  the combinatorial identity
$$
{n+\alpha\choose n-k}=(-1)^k{(\alpha+1)_n(-n)_k\over n!(\alpha+1)_k}
$$
we have the following

\begin{prop} The explicit representation for
the RJP $P_n^{(\alpha,\beta;N)}(\rho(N,\beta);x)$
holds true:
$$
\displaylines{
\hspace{-5cm} P_n^{(\alpha,\beta;N)}(\rho(N,\beta);x):=\cr
\noalign{\vskip3truemm}
={(\alpha+1)_n\over n!}\ \sum^n_{k=0}{1\over k!}\
{(-n)_k\left[{[\alpha+(1+\rho(N,\beta))\beta+n+1]N+\left(n-{1\over
2}\right)\beta\over N-\beta}\right]_k\over
(\alpha+1)_k}\left({(\beta-N)(x-1)\over 2N}\right)^k=\cr
={(\alpha+1)_n\over
n!}{}_{2}F_1\left[-n,{[\alpha+(1+\rho(N,\beta))\beta+n+1]N+\left(n-{1\over
2}\right)\beta\over N-\beta},\alpha+1;{(\beta-N)(x-1)\over
2N}\right]
\cr}
$$\end{prop}

\begin{prop} The RJP
$P_n^{(\alpha,\beta;N)}(\rho(N,\beta); x)$ are
solutions of the hypergeometric differential equation:
\be\label{4.11}
\sigma(x,N){\d^2y_n\over\d x^2}+\tau(x;n;N){\d y_n\over \d x}+
\lambda_n(N)y_n=0\ ,
\ee
where
\begin{eqnarray*}
\sigma(x;N)&=&\left[{N+\beta\over N-\beta}-{2\beta\over
N-\beta}x-x^2\right]\ ,\\
\noalign{\vskip6truemm}
\tau(x;n;N)&=&{[(1+\rho(N,\beta))\beta-\alpha]N+\left(2n-{3\over2}\right)
\beta\over{N -\beta}} - {[\alpha+(1+\rho(N,\beta))\beta+2]N+\left(2n-{3\over
2}\right)\beta\over N-\beta}\ x\ ,\\
\noalign{\vskip4truemm}
\lambda_n(N)&=&{n\over
N-\beta}\left[[\alpha+(1+\rho(N,\beta))\beta+n+1]N+\left(n-{1\over
2}\right)\beta\right]\ .
\end{eqnarray*}
\end{prop}

\begin{prop}  The symmetrization factor
for eq.  {\rm (\ref{4.11})} is given by
$$
\mu_n(x;N;\rho(N,\beta)):=(1-x)^{\alpha}\left(x+{N+\beta\over
N-\beta}\right)^{\beta[(1+\rho(N,\beta))N+2n+{1\over
2}+\alpha]\over N-\beta}
$$
so that, if $N>\beta$, the support of the orthogonality measure is 
\be\label{4.12}
I:=\left[-{N+\beta\over N-\beta},1\right]\ .
\ee
\end{prop}

\begin{prop}  The RJP
$P_n^{(\alpha,\beta;N)}(\rho(N,\beta); x)$ can be
defined by the Rodrigues formula:
$$
\displaylines{
P_n^{(\alpha,\beta;N)}(\rho(N,\beta);x):=\cr
\noalign{\vskip3truemm}
={(-1)^n\over 2^n n!}\left({N-\beta\over
N}\right)^n(1-x)^{-\alpha}\left(x+{N+\beta\over
N-\beta}\right)^{-{\beta[(1+\rho(N,\beta))N+2n+{1\over
2}+\alpha]\over N-\beta}}\times\cr
\times{\d^n\over\d
x^n}\left\{(1-x)^{\alpha+n}\left(x+{N+\beta\over
N-\beta}\right)^{{\beta[(1+\rho(N,\beta))N+2n+{1\over
2}+\alpha]\over N-\beta}+n}\right\}\cr}
$$
\end{prop}

\begin{prop}  The RJP
$P_n^{(\alpha,\beta;N)}(\rho(N,\beta); x)$ verify
the three-term recurrence relation:
$$
\displaylines{
\hskip-3.2truecm(4.13)\hskip3.5truecm
A_{n,N}P_{n+1}^{(\alpha,\beta;N)}(\rho(N,\beta);x)=\cr
\noalign{\vskip3truemm}
=[B_{n,N}-x]P_n^{(\alpha,\beta;N)}(\rho(N,\beta);x)+
C_{n,N}P_{n-1}^{(\alpha,\beta;N)} (\rho(N,\beta);x)\ ,\cr}
$$
$$
\left(P_{-1}^{(\alpha,\beta;N)}(\rho(N,\beta);x):=0\
;\quad
P_0^{(\alpha,\beta;N)}(\psi(\alpha);\rho(N,\beta);x):=1\right)\
,
$$
where:
$$
A_{n,N}=(n+1){2N\over
N-\beta}{\left[{[\alpha+(1+\rho(N,\beta))\beta+n+1]N+
\left(n-{1\over 2}\right)\beta\over N-\beta}\right]_n\over
\left[{[\alpha+(1+\rho(N,\beta))\beta+n+2]N+\left(n+{1\over
2}\right)\beta\over N-\beta}\right]_{n+1}}
$$
\begin{eqnarray*}
B_{n,N}&={\displaystyle1+{2nN(\alpha+n)\over
N[\alpha+(1+\rho(N,\beta))\beta+2n]+{1\over
2}\beta}}\\
\noalign{\vskip4truemm}
&{\displaystyle +{2N(n+1)(\alpha+n+1)\over
N[\alpha+(1+\rho(N,\beta))\beta+2n+2]+{1\over
2}\beta}}\end{eqnarray*}
$$
\displaylines{
C_{n,N}=(n+\alpha)\left({2N\over
N-\beta}\right)^{N+\beta\over N-\beta}\
{N[\alpha+(1+\rho(N,\beta))\beta+2n-1]-{1\over
2}\beta\over
N[\alpha+(1+\rho(N,\beta))\beta+2n+1]-{1\over
2}\beta}\times\cr
\noalign{\vskip4truemm}
\times
{\left[{[(1+\rho(N,\beta))\beta+n]N+\left(\alpha+n-{3\over2}\right)\beta\over
N-\beta}\right]_{\alpha+n-1}\over
\left[{[(1+\rho(N,\beta))\beta+n+1]N+\left(\alpha+n-{1\over
2}\right)\beta\over N-\beta}\right]_{\alpha+n}}\cr}
$$\end{prop}

\begin{prop} The RJP
$P_n^{(\alpha,\beta;N)}(\rho(N,\beta); x)$ satisfy
the varying orthogonality property:
$$
\int_IP_n^{(\alpha,\beta;N)}(\rho(N,\beta); x)
P_m^{(\alpha,\beta;N)}(\rho(N,\beta);
x)\mu^{(N)}_{n,m}(x)\d x=k_n\delta_{n,m}\ ,
$$
where $I$ is defined by {\rm (\ref{4.12})}, $k_n$ denotes the square of the
weighted $L^2$ norm of the $n$-th polynomial
$P_n^{(\alpha,\beta;N)}(\rho(N,\beta); x)$, which is
obviously depending on  $\rho(N,\beta)$,
$$
\displaylines{
\mu^{(N)}_{n,m}(x)=(1-x)^{\alpha}\left(x+{N+\beta\over
N-\beta}\right)^{\beta[(1+\rho(N,\beta))N+2n+2m+{1\over
2}+\alpha]\over N-\beta}=\cr
=\varphi_{n,N}(x)\varphi_{m,N}(x)(1-x)^\alpha
\left(x+{N+\beta\over
N-\beta}\right)^{\beta[(1+\rho(N,\beta))N+{1\over
2}+\alpha]\over N-\beta}\ ,\cr}
$$
so that, by putting
$$
\varphi_{n,N}(x):=\left(x+{N+\beta\over
N-\beta}\right)^{2n\beta\over N-\beta}\ ,
$$
$$
\mu^{(\alpha,\beta;N)}(\rho(N,\beta);x):=(1-x)^{\alpha}
\left(x+{N+\beta\over N-\beta}\right)^{\beta[(1+\rho(N,\beta))N+{1\over
2}+\alpha]\over N-\beta}\ ,
$$
the complete orthogonality condition can be written in the form
{\rm (\ref{3.7})}:
$$
\int_I\left[\varphi_{n,N}P_n^{(\alpha,\beta;N)}\right]\
\left[\varphi_{m,N}P_m^{(\alpha,\beta;N)}\right]\mu^{(\alpha,\beta;N)}\d
x=k_n\delta_{n,m} \ .
$$\end{prop}

\section*{Conclusion}

 The extremely wide possibility to construct Relativistic
Jacobi Polynomials, simply by choosing in the preceding section function
 $\rho(N,\beta)$ satisfying the prescribed limit conditions, seems to
give relevant importance to analyze the physical meaning of such sets of
polynomials orthogonal with respect to varying weights.

\section*{Note added in proofs}

In a recent paper Prof. Mourad E.H. ISMAIL (J. Phys. A: Math. Gen. {\bf 29} 
(1996), 3199-3202) expressed a severe criticism about relativistic
orthogonal polynomials, since they can be obtained from the classical 
Jacobi polynomials by means of suitable changes of variable.
We want to remind that in every paper on this subject we have pointed 
out the existence of a Nagel type formula, expressing relativistic 
polynomials in terms of the Gauss hypergeometric function.   However, 
in our opinion, the physical meaning of the relativistic Hermite 
polynomials and the possibility to construct families of polynomials
(depending on $N$) related to each classical weight function deserve
a particular attention.   It was not probably clear, {\it a priori},
how to modify the classical Jacobi polynomials, by suitable changes
of variable, in order to obtain the considered polynomial families.


\section*{References}

\begin{description}
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quantum relativistic harmonic oscillator: The relativistic Hermite
polynomials}, {\sc Phys. Lett. A}, {\bf 156} (1991), 381-385.
\item{[2]} {\sf C. Belingeri and P.E. Ricci}: {\it General Sets of Relativistic
Jacobi Polynomials}, (submitted).
\item{[3]} {\sf B. Germano, S. Noschese, and P.E. Ricci}: {\it
General Sets of Relativistic Laguerre Polynomials},  (submitted).
\item{[4]} {\sf M. He, S. Noschese, and P.E. Ricci}: {\it The Relativistic
Szeg\"o Polynomials}, {\sc Integral Transf. and Spec. Funct.} (in print).
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Polynomials}, {\sc Integral Transf. and Spec. Funct.} (in print).
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zeros of Relativistic Hermite Polynomials}, {\sc SIAM J. Math. Anal.} (in
print).
\item{[7]} {\sf M. He and K. Pan}: {\it Distribution of the Zeros of
Relativistic Jacobi Polynomials},  (submitted).
\item{[8]} {\sf B. Nagel}: {\it The relativistic Hermite polynomial
is a Gegenbauer polynomial}, {\sc J. Math. Phys.}, {\bf 35} (1994), 1549-1554.
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Polynomials}, {\sc Rend. Mat. Appl.}, S. VII, {\bf 16} (1996), 299-313.
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Relativistic Laguerre Polynomials}, {\sc Atti Sem. Mat. Fis. Univ. Modena}
(in print).
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the Density of Zeros for the Relativistic Jacobi Polynomials}, {\sc Rend.
Ist. Mat. Univ. Trieste} (in print).
\item{[13]} {\sf P. Natalini and S. Noschese}: {\it The Relativistic
Bessel Polynomials}, {\sc D.S. Mitrinovic Memorial Volume} (in print).
\item{[14]} {\sf P. Natalini, S. Noschese and P.E. Ricci}: {\it On the
moments of the density of zeros for the Relativistic Bessel
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\item{[15]} {\sf A.F. Nikiforov and V.B. Uvarov}: {\bf Special
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\item{[16]} {\sf S. Noschese}: {\it A Vyskov-type formula for the
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(in print).
\item{[17]} {\sf P.E. Ricci and S. Noschese}: {\it On the moments of
the density of zeros for the Relativistic Hermite and Laguerre
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\item{[18]} {\sf A. Zarzo and A. Martinez}: {\it The
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\item{[19]} {\sf A. Zarzo and A. Martinez}: {\it Varying
orthogonality for a class of hypergeometric type polynomials},
{\sc II International Seminar on Approximation and Optimization}, La
Habana, Cuba, Sept. 1993. Approximation and Optimization series,
Peter Lang Verlag, Berlin (1994) (in print).

\item{[20]} {\sf A. Zarzo, J.S. Dehesa, and A. Martinez}: {\it On a new
set of polynomials representing the wave functions of the
quantum relativistic harmonic oscillator}, {\sc J. Phys. A: Math. Gen.} (in
print).
\item{[21]} {\sf A. Zarzo}: {\it Ecuaciones Diferenciales de Tipo
Hipergeom\'etrico}, {\sc Tesis Doctoral}, Universidad de Granada,  Spain, 1995.
\end{description}
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{\sc 
{\obeylines
Pierpaolo NATALINI - Paolo Emilio RICCI
Dipartimento di Matematica ``Guido CASTELNUOVO''
Universit\`a degli Studi di Roma ``La Sapienza''
P.le A. Moro, 2
00185 -- ROMA 
{\small \tt {\sf  E-mail:} RICCIP@SCI.UNIROMA1.IT}
Fax \#: (396) - 44701007}

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\obeylines{
{Matthew Xisheng HE}
Nova Southeastern University - Farquhar Center
3301 College Avenue
Fort Lauderdale - Florida 33314
(U.S.A.)
{\small \tt {\sf E-mail}: HEM@POLARIS.NCS.NOVA.EDU} }
}

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