# The symmetric Dunkl-classical orthogonal polynomials revisited

Khalfa DOUAK111Email<EMAIL_ADDRESS>
Laboratoire Jacques-Louis Lions, Sorbonne Université,
75252 Paris Cedex 05, France
In memory of Pascal MARONI

###### Abstract

We investigate the symmetric Dunkl-classical orthogonal polynomials by using a
new approach applied in connection with the Dunkl operator. The main aim of
this technique is to determine the recurrence coefficients first and foremost.
We establish the existence and uniqueness of symmetric Dunkl-classical
orthogonal polynomials, and also confirm that only two families of orthogonal
polynomials, that is, the generalized Hermite polynomials and the generalized
Gegenbauer polynomials, belong to this class. Two apparently new
characterizations in a more general setting are given. This paper complements
earlier work of Ben Cheikh and Gaied.

Keywords. Dunkl Operator, orthogonal polynomials, generalized Hermite
polynomials, generalized Gegenbauer polynomials, recurrence relations,
differential equations, difference equations.
AMS Classification. 33C45; 42C05.

##  1 Introduction and motivation

The original Dunkl operators as called in the literature are a family of
first-order differential-difference operators first introduced by Charles
Dunkl in [12] whose the main idea was to use finite reflection groups to
provide structure for multi-variable analysis. Here we are concerned with the
one-dimensional Dunkl operator, denoted by $T_{\mu}$, defined as follows

$\displaystyle\big{(}T_{\mu}f\big{)}(x)=f^{\prime}(x)+\mu\frac{f(x)-f(-x)}{x},$
(1.1)

where $f$ is a polynomial or an entire function and $\mu$ is taken here to be
an arbitrary real parameter.
Acting on polynomials, one sees at once that the Dunkl operator $T_{\mu}$
reduces the degree of every polynomial by exactly one. In particular, if we
take $f(x)=x^{n}$, we get $T_{\mu}x^{n}=\mu_{n}x^{n-1}$ where
$\mu_{n}=n+\mu\left(1-(-1)^{n}\right),\ n\geqslant 0$, with
$\mu\neq-n-\frac{1}{2},\ n\geqslant 0$.
We are interested here in the problem of finding the sequences of symmetric
orthogonal polynomials $\\{P_{n}\\}_{n\geqslant 0}$ when the sequence of
polynomials $\\{T_{\mu}P_{n}\\}_{n\geqslant 0}$ is also orthogonal.
In an earlier work [3], Ben Cheikh and Gaied considered this problem and
referred to the resulting polynomials as the symmetric Dunkl-classical
orthogonal polynomials. In doing so, the authors have focused their study on
the identification of solutions of a differential-difference equation of type

$\displaystyle\sigma(x)T_{\mu}^{2}P_{n}(x)+\tau(x)T_{\mu}P_{n}(x)+\lambda_{n}P_{n}(x)=0,$
(1.2)

where $\sigma(x)$ and $\tau(x)$ are two polynomials independent to $n$,
$\sigma(x)$ is an even polynomial of degree less than or equal to two,
$\sigma(x)$ is an odd polynomial of first degree and $\lambda_{n}$ is
independent of $x$. They deduced that the generalized Hermite polynomials and
the generalized Gegenbauer polynomials are the only eigenfunctions of (1.2).

In this paper we investigate this kind of polynomials by adopting a different
approach. This technique first used in my Ph.D. thesis [9] to construct the
classical orthogonal polynomials, part of which was published in [10]. It was
also applied later in many other works, for instance, to construct the
families of $D_{\omega}$-classical orthogonal polynomials [11]. To continue in
this line, it is then reasonable to ask whether this can be extended to the
Dunkl context, where the ordinary derivative $D$ (or $D_{\omega}$) is replaced
by the Dunkl operator $T_{\mu}$. This allows us to construct the symmetric
Dunkl-classical orthogonal polynomials. Unfortunately, this method is rather
complicated, if not impossible, in the general case. Therefore our interest is
in describing such orthogonal polynomials only for the symmetric case.
Roughly speaking, the main idea behind this approach is to focus first on the
explicit determination of recurrence coefficients. This in fact allows us to
establish a link between the Dunkl-classical character and the form of the
recurrence coefficients which represents a new result characterizing the
symmetric Dunkl-classical orthogonal polynomials (Theorem 2.1). As an
important consequence of this theorem we also bring another result
characterizing the resulting polynomials via a differential-difference
equation in a more general setting than that stated in (1.2) (Theorem 2.2).
As an application of Theorem 2.1, we can assert that the generalized Hermite
polynomials and the generalized Gegenbauer polynomials are the only symmetric
Dunkl-classical orthogonal polynomials. Subsequently, we sharpen the results
stated in Theorem 2.2 to characterize each of the above two families of
polynomials by means of a differential-difference equation of type (1.2).
Before we can state the main theorems in this paper we wish to introduce some
standard notations.
Let $\mathscr{P}$ be the vector space of polynomials of one variable with
complex coefficients and $\mathscr{P}^{\prime}$ its algebraic dual. By
$\\{P_{n}\\}_{n\geqslant 0}$ we denote a polynomials sequence (PS in short),
with $\deg P_{n}=n$, and $\\{u_{n}\\}_{n\geqslant 0}$ its associated dual
sequence defined by $\bigl{<}u_{n},P_{m}\bigr{>}=\delta_{nm};\ n,m\geqslant
0,$ where $\delta_{nm}$ is the Kronecker’s delta symbol. Throughout this
article, we will always consider the sequence of monic polynomials, i.e. the
leading coefficient of each polynomial $P_{n}$ is one
($P_{n}(x)=x^{n}+\cdots$). We also recall the two linear operators:
The ordinary differential operator $D=d/dx$ :
$\left(Df\right)(x)=f^{\prime}(x)$.
The Hahn operator $\,H_{q,\omega}$ defined for two fixed complex numbers $q$
and $\omega$ by

$\left(H_{q,\omega}f\right)(x):=\frac{f(qx+\omega)-f(x)}{(q-1)x+\omega},\quad\forall
f\in\mathscr{P}.$ (1.3)

When $q=1$ with $\omega\neq 0$ we get the discrete operator
$\left(D_{\omega}f\right)(x):=\left(H_{1,\omega}f\right)(x)$, that is,

$\left(D_{\omega}f\right)(x)=\frac{f(x+\omega)-f(x)}{\omega}.$ (1.4)

For $q\neq 1$ and $\omega=0$, the $q$-difference operator, which we write
$\left({H}_{q}f\right)(x):=\left(H_{q,0}f\right)(x)$, is given by

$\left(H_{q}f\right)(x):=\frac{f(qx)-f(x)}{(q-1)x}.$ (1.5)

With this notation, we can write (1.1) in the form

$\displaystyle\big{(}T_{\mu}f\big{)}(x)=\left(Df\right)(x)+2\mu\left(H_{-1}f\right)(x).$
(1.6)

To begin with, we consider a sequence of monic orthogonal polynomials (MOPS)
with its associated linear functional denoted $u\in\mathscr{P}^{\prime}$. The
sequence of complex numbers $(u)_{n},\ n=0,1,2,\ldots,$ denotes the moments of
$u$ with respect to the sequence $\\{x^{n}\\}_{n\geqslant 0}$, namely, the
moment of order $n$ for the functional $u$ is denoted by
$(u)_{n}:=\big{<}u,x^{n}\big{>},\ n=0,1,2,\ldots$. Thus, the linear functional
$u$ is completely determined by its moments. To terminate we define by
$\Delta_{n}=\det\big{(}(u)_{i+j}\big{)}_{i,j=0}^{n},\ n\geqslant 0$, the
ordinary Hankel determinants constructed from the moments of $u$.

###### Definition 1.1

A PS $\\{P_{n}\\}_{n\geqslant 0}$ is said to be orthogonal with respect to
(w.r.t.) the linear functional $u$, if it satisfies the orthogonality
conditions

$\displaystyle\big{<}u\,,\,P_{n}P_{m}\big{>}$ $\displaystyle=0,\ n\neq m,$
(1.7a) $\displaystyle\big{<}u\,,\,P_{n}^{2}\big{>}$ $\displaystyle\not=0,\
n\geqslant 0.$ (1.7b)

In this case (1.7b) are said to be the regularity conditions. As an immediate
consequence of the regularity of $u$, we have $(u)_{0}\neq 0$ and $u=\lambda
u_{0}$ with $\lambda\neq 0$. For this reason, we always consider the
orthogonality w.r.t. $u_{0}$ which is the canonical form associated to the PS
$\\{P_{n}\\}_{n\geqslant 0}$.

###### Theorem 1.2

[7] A necessary and sufficient condition for the existence of an orthogonal
polynomials sequence (OPS) $\\{P_{n}\\}_{n\geqslant 0}$ with respect to
$u_{0}$ is $\Delta_{n}=\det\big{(}(u_{0})_{i+j}\big{)}_{i,j=0}^{n}\neq 0$, for
$n=0,1,2,\ldots$. The functional $u_{0}$ is then said to be regular.

Equivalently, the polynomials $P_{n},\ n=0,1,2,\ldots$, are orthogonal if and
only if they satisfy the well-known second-order recurrence relation

$\displaystyle P_{n+2}(x)=(x-\beta_{n+1})P_{n+1}(x)-\gamma_{n+1}P_{n}(x),\
n\geqslant 0,$ (1.8a) $\displaystyle P_{0}(x)=1,\ P_{1}(x)=x-\beta_{0},$
(1.8b)

with the regularity conditions $\gamma_{n+1}\neq 0,\ n\geqslant 0$.
The linear functional $u$ is said to be regular if there exists a sequence of
polynomials $\\{P_{n}\\}_{n\geqslant 0}$, such that
1. $\deg P_{n}=n$, $\forall n\in\mathbb{N}$,
2. $\big{<}u\,,\,P_{n}P_{m}\big{>}=k_{n}\delta_{nm},\ k_{n}\neq 0,\ \forall n,m\in\mathbb{N}$. If $k_{n}>0$, for all $n\in\mathbb{N}$, then $u$ is positive definite.

Note that we work here with linear functionals which we assume to be regular,
but not necessarily positive definite.
A regular linear functional $u$ is said to be semi-classical if there exist
two polynomials $\phi$ and $\psi$, with $\deg\phi\geqslant 0$ and
$\deg\psi\geqslant 1$, satisfying [19]

$\displaystyle D\left(\phi(x)u\right)+\psi(x)u=0.$ (1.9)

If moreover $\phi(x)=a_{t}x^{t}+...$, and $\psi(x)=b_{p}x^{p}+...$, then one
of the following conditions holds:
$\bf a$. $t-1\neq p$,
$\bf b$. if $t-1=p$, then $na_{t}\neq b_{p},\forall n\in\mathbb{N}$, and the
class of $u$ is defined as

$\displaystyle s=\max\left(\deg\phi-2,\deg\psi-1\right).$ (1.10)

Under these conditions, the pair $(\phi,\psi)$ is said admissible.

Define now the sequence $\\{Q_{n}\\}_{n\geqslant 0}$, where
$Q_{n}(x)=\mu_{n+1}^{-1}\big{(}T_{\mu}P_{n+1}\big{)}(x),\,n\geqslant 0$. This
normalization plays an important role in what follows and its choice seems to
be the best adapted to our approach. The classical orthogonal polynomial
sequences of Hermite, Laguerre, Jacobi and Bessel collectively satisfy the so-
called Hahn property [14]: the sequence of its derivatives is again an
orthogonal polynomial sequence. Likewise, we can introduce the adjective
Dunkl-classical or simply “$T_{\mu}$-classical” of an OPS as follows.

###### Definition 1.3

The OPS $\\{P_{n}\\}_{n\geqslant 0}$ is called “$T_{\mu}$-classical” if the
sequence $\\{Q_{n}\\}_{n\geqslant 0}$ is also a OPS.

The $\\{P_{n}\\}_{n\geqslant 0}$ is said to be symmetric if each polynomial
fulfills $P_{n}(-x)=(-1)^{n}P_{n}(x),\,n\geqslant 0$. Further, when
$\\{P_{n}\\}_{n\geqslant 0}$ is an OPS, all of the moments of the odd order of
the form $u_{0}$ are zero.

###### Theorem 1.4

[7] For every MOPS $\\{P_{n}\\}_{n\geqslant 0}$, the following statements are
equivalent:
(1) The form $u_{0}$ is symmetric.
(2) The sequence $\\{P_{n}\\}_{n\geqslant 0}$ is symmetric.
(3) The sequence $\\{P_{n}\\}_{n\geqslant 0}$ satisfies the recurrence
relation

$\displaystyle P_{n+2}(x)=xP_{n+1}(x)-\gamma_{n+1}P_{n}(x),\ n\geqslant 0,$
(1.11a) $\displaystyle P_{0}(x)=1,\ P_{1}(x)=x.$ (1.11b)

The following identity is straightforward:

$\displaystyle\big{(}T_{\mu}fg\big{)}(x)=f\big{(}T_{\mu}g\big{)}(x)+g\big{(}T_{\mu}f\big{)}(x)-4\mu
x\left(H_{-1}f\right)(x)\left(H_{-1}g\right)(x).$ (1.12)

Suppose now that $\\{P_{n}\\}_{n\geqslant 0}$ is “$T_{\mu}$-classical”. Thus
$\\{Q_{n}\\}_{n\geqslant 0}$ is also orthogonal and then satisfies a second
order recurrence relation. Furthermore, if we assume that
$\\{P_{n}\\}_{n\geqslant 0}$ is symmetric, we easily check that
$\\{Q_{n}\\}_{n\geqslant 0}$ is also symmetric, that is,

$\displaystyle Q_{n+2}(x)=xQ_{n+1}(x)-{\tilde{\gamma}}_{n+1}Q_{n}(x),\
n\geqslant 0,$ (1.13a) $\displaystyle Q_{0}(x)=1,\ Q_{1}(x)=x.$ (1.13b)

As mentioned above, it is not so easy to apply our procedure when the PS
$\\{P_{n}\\}_{n\geqslant 0}$ is not symmetric. Accordingly we are in effect
dealing only with the symmetric case and therefore our starting points are the
recurrence relations (1.11a)-(1.11b) and (1.13a)-(1.13b).

First, from (1.11a)-(1.11b) by applying the operator $T_{\mu}$ we get

$\displaystyle\big{(}T_{\mu}P_{n+2}\big{)}(x)=\big{(}T_{\mu}xP_{n+1}\big{)}(x)-\gamma_{n+1}\big{(}T_{\mu}P_{n}\big{)}(x),\
n\geqslant 0,$ (1.14a) $\displaystyle\big{(}T_{\mu}P_{0}\big{)}(x)=0,\
\big{(}T_{\mu}P_{1}\big{)}(x)=\mu_{1}.$ (1.14b)

Applying the formula (1.12) for $f(x)=x$ and $g(x)=P_{n+1}(x)$, we see that

$\displaystyle\big{(}T_{\mu}xP_{n+1}\big{)}(x)=x\big{(}T_{\mu}P_{n+1}\big{)}(x)+\xi_{n+1}P_{n+1},\
n\geqslant 0,$ (1.15)

where we have put $\xi_{n}=1+2\mu(-1)^{n},\ n\geqslant 0$. Substituting (1.15)
into (1.14a) yields

$\displaystyle\xi_{n+1}P_{n+1}(x)=\mu_{n+2}Q_{n+1}(x)+\mu_{n}\gamma_{n+1}Q_{n-1}(x)-\mu_{n+1}xQ_{n}(x),\
n\geqslant 0.$ (1.16)

Using the $\xi_{n}$-coefficients will be very helpful later on to simplify
most operations in the next two sections. As one can easily see, the
coefficients $\xi_{n}$ and $\mu_{n}$ are interrelated via

$\displaystyle\xi_{n}=\mu_{n+1}-\mu_{n},\ n\geqslant 0.$ (1.17)

At the end of this section, let us remember the definition of the shifted
polynomials denoted $\\{\hat{P}_{n}\\}_{n\geqslant 0}$ corresponding to the PS
$\\{P_{n}\\}_{n\geqslant 0}$. For all $n,n=0,1,\ldots,$ we have

$\displaystyle{\hat{P}}_{n}(x):=a^{-n}P_{n}({a}x+{b}),\ \hbox{\rm for}\
({a};{b})\in\mathbb{C}^{\ast}\times\mathbb{C}.$ (1.18)

Since the classical character of the considered polynomials is preserved by
any linear change of the variable, for the OPS $\\{P_{n}\\}_{n\geqslant 0}$
satisfying (1.8a)-(1.8b), we obtain that the polynomials
${\hat{P}}_{n},{n=0,1,\ldots}$, satisfy also the second-order recurrence
relation

$\displaystyle{{\hat{P}}}_{n+2}(x)=(x-{\hat{\beta}}_{n+1}){{\hat{P}}}_{n+1}(x)-{\hat{\gamma}}_{n+1}{{\hat{P}}}_{n}(x),\
n\geqslant 0,$ (1.19a) $\displaystyle{{\hat{P}}}_{1}(x)=x-{\hat{\beta}}_{0},\
{{\hat{P}}}_{0}(x)=1,$ (1.19b)

with

$\displaystyle{\hat{\beta}}_{n}=\frac{{\beta}_{n}-{b}}{{a}},\,n\geqslant 0,\ \
\mbox{and}\ \ {\hat{\gamma}}_{n+1}=\frac{{\gamma}_{n+1}}{{a}^{2}},\,n\geqslant
0\quad({a}\neq 0).$ (1.20)

Now, under the hypothesis that $\\{P_{n}\\}_{n\geqslant 0}$ is symmetric, we
have ${\hat{\beta}}_{n}={\beta}_{n}=0,\,n\geqslant 0$.

The structure of this paper is as follows. In Section 2, we see how Definition
1.3 allows us to construct the symmetric $T_{\mu}$-classical orthogonal
polynomials by solving a system of nonlinear equations satisfied by the
coefficients ${\gamma}_{n}$ and ${\hat{\gamma}}_{n}$. Based on solutions of
this system we state and prove the two theorems characterizing the symmetric
Dunkl-classical orthogonal polynomials. The third section is devoted to the
application of both theorems. Theorem 2.1 asserts that the only symmetric
Dunkl-classical orthogonal polynomials are the generalized Hermite polynomials
and the generalized Gegenbauer polynomials, and Theorem 2.2 gives a
characterization of these families via a differential-difference equation of
type (1.2). Many other known properties of each family are given.

##  2 Construction of the symmetric $T_{\mu}$-classical orthogonal polynomials

The main result of the present paper is the following theorem which ensures
the existence and the uniqueness of the symmetric Dunkl-classical orthogonal
polynomial sequences. For this purpose, we show that the coefficients
${\gamma_{n}}$ have a specific rational expression which actually
characterizes the symmetric Dunkl-classical orthogonal polynomials.

###### Theorem 2.1

Let $\\{P_{n}\\}_{n\geqslant 0}$ be a MOPS and $\\{Q_{n}\\}_{n\geqslant 0}$
the sequence defined by
$Q_{n}(x)=\mu_{n+1}^{-1}\big{(}T_{\mu}P_{n+1}\big{)}(x)$. Then
$\\{P_{n}\\}_{n\geqslant 0}$ satisfies the recurrence relation (1.11a)-(1.11b)
with

$\displaystyle{\gamma_{n+2}}=\frac{\xi_{n+2}}{\xi_{n+1}}\frac{\mu_{n+2}}{\mu_{n+1}}\frac{\big{(}\mu_{n}\theta_{n}-\mu_{n-1}\big{)}}{\big{(}\mu_{n+3}\theta_{n+1}-\mu_{n+2}\big{)}}{\gamma_{n+1}},\
n\geqslant 0,$ (2.1)

where $\theta_{n}$ a solution to the Riccati equation

$\displaystyle\xi_{n+1}\theta_{n+1}+{\xi_{n}}{\theta_{n}^{-1}}$
$\displaystyle=2,\ n\geqslant 1,$ (2.2)

if and only if $\\{Q_{n}\\}_{n\geqslant 0}$ is a MOPS satisfying
(1.13a)-(1.13b) with

$\displaystyle\tilde{\gamma}_{n}=\frac{\mu_{n}}{\mu_{n+1}}\theta_{n}\gamma_{n+1},\
n\geqslant 1.$ (2.3)

Note that, in the identity (2.1), we have adopted the conventions that
$\theta_{0}=1$ and $\mu_{-1}=-1+2\mu$.
Proof First we shall prove that if $\\{Q_{n}\\}_{n\geqslant 0}$ is a MOPS
satisfying (1.13a)-(1.13b) with (2.3), then $\\{P_{n}\\}_{n\geqslant 0}$
satisfies (1.11a)-(1.11b) with (2.1)-(2.2).
On account of (1.13a)-(1.13b) with the use of (2.3), it is easily seen that
the equation (1.16) writes

$\displaystyle P_{n+1}(x)$
$\displaystyle=Q_{n+1}(x)+\frac{\mu_{n}}{\xi_{n+1}}\gamma_{n+1}\big{(}1-\theta_{n}\big{)}Q_{n-1}(x),\
n\geqslant 1,$ (2.4a) $\displaystyle P_{1}(x)$
$\displaystyle=Q_{1}(x)=x,\,P_{0}(x)=Q_{0}(x)=1.$ (2.4b)

Substituting, in (1.11a), $P_{n+2},P_{n+1}$ and $P_{n}$ by their expressions
provided in (2.4a), using again (1.13a) we get an expansion in terms of
$Q_{n}$ and $Q_{n-2}$. After an easy computation we derive the equations

$\displaystyle\frac{\big{(}\mu_{n+3}\theta_{n+1}-\mu_{n+2}\big{)}}{\xi_{n+2}}\frac{\gamma_{n+2}}{\mu_{n+2}}$
$\displaystyle=\frac{\big{(}\mu_{n}\theta_{n}-\mu_{n-1}\big{)}}{\xi_{n+1}}\frac{\gamma_{n+1}}{\mu_{n+1}},\
n\geqslant 1,$
$\displaystyle\big{(}\mu_{3}\theta_{1}-\mu_{2}\big{)}\gamma_{2}$
$\displaystyle=\mu_{2}\gamma_{1};$
$\displaystyle\xi_{n+1}\theta_{n+1}+{\xi_{n}}{\theta_{n}^{-1}}$
$\displaystyle=2,\ n\geqslant 1.$

Observe that the last equation is exactly (2.2). If moreover we adhere to the
above conventions, the first two ones may be grouped together as in (2.1). The
condition is then sufficient.
Conversely, if now $\\{P_{n}\\}_{n\geqslant 0}$ fulfills (1.11a)-(1.11b) with
$\gamma_{n+1}$ and $\theta_{n}$ satisfying (2.1) and (2.2), we will show that
$\\{Q_{n}\\}_{n\geqslant 0}$ is a symmetrical MOPS. In this part, the proof
requires tedious computations and its basic idea is similar in spirit to that
of [17, Th. 3.4].
Changing $n$ into $n-1$ in (1.11a), then substitute $P_{n+1}$, $P_{n}$ and
$P_{n-1}$ by their respective expressions provided in (1.16) we obtain

$\displaystyle{\xi_{n}}{\mu_{n+2}}Q_{n+1}(x)$
$\displaystyle=2{\mu_{n+1}}xQ_{n}(x)+2{\mu_{n-1}}\gamma_{n}xQ_{n-2}(x)-\xi_{n+1}{\mu_{n}}x^{2}Q_{n-1}(x)$
$\displaystyle\
-{\xi_{n}}\mu_{n}\left(\gamma_{n+1}+\gamma_{n}\right)Q_{n-1}(x)-{\xi_{n}}\mu_{n-2}\gamma_{n}\gamma_{n-1}Q_{n-3}(x)\quad(Q_{-n}=0,\,n\geqslant
1).$ (2.5)

As the sequence $\\{Q_{n}\\}_{n\geqslant 0}$ is a basis for the vector space
$\mathscr{P}$, there exist a coefficients $\lambda_{n+1,\nu}$ such that

$\displaystyle
xQ_{n}(x)=\sum_{\nu=0}^{n+1}\lambda_{n+1,\nu}Q_{\nu}(x),\,n\geqslant 0,$ (2.6)

where $\lambda_{n+1,n+1}=1$ because $Q_{n}$ is monic (with
$\lambda_{n+1,\nu}=0$ if $\nu<0$ or $\nu>n+1$).
The objective here is to show that $\lambda_{n+1,n-1}\neq 0,\,n\geqslant 1$,
and, for all $\nu\neq n-1$, we have $\lambda_{n+1,\nu}=0$.
Use of (2.6) shows that

$\displaystyle
x^{2}Q_{n}(x)=\sum_{\nu=1}^{n+2}\sum_{\tau=\nu-1}^{n+1}\lambda_{n+1,\tau}\lambda_{\tau+1,\nu}Q_{\nu}(x)+\sum_{\tau=0}^{n+1}\lambda_{n+1,\tau}\lambda_{\tau+1,0}Q_{0}(x),\,n\geqslant
0.$ (2.7)

Because of the symmetry property of $\\{Q_{n}\\}_{n\geqslant 0}$, due to
(2.6), it is easy to check that the coefficients $\lambda_{n+1,n-\nu}=0$, for
each even number $\nu$, $\ 0\leqslant\nu\leqslant n$. In other words,
$\lambda_{n,\nu}=0$ whenever $n-\nu\equiv 1\ \mbox{\rm mod}\ 2.$
Thus, to prove (1.13a)-(1.13b), it only remains to verify that
$\lambda_{n+1,n-\nu}=0$, for each odd number $\nu\neq 1$.
For this, replace in (2.5) the terms $xQ_{n}$, $xQ_{n-2}$ and $x^{2}Q_{n-1}$
by their respective expressions obtained from (2.6) and (2.7) to get

$\displaystyle{\xi_{n}}{\mu_{n+2}}Q_{n+1}(x)$
$\displaystyle=2{\mu_{n+1}}\sum_{\nu=0}^{n+1}\lambda_{n+1,\nu}Q_{\nu}(x)+2{\mu_{n-1}}\gamma_{n}\sum_{\nu=0}^{n-1}\lambda_{n-1,\nu}Q_{\nu}(x)$
$\displaystyle\
\,-\xi_{n+1}{\mu_{n}}\Big{[}\sum_{\nu=1}^{n+1}\sum_{\tau=\nu-1}^{n}\lambda_{n,\tau}\lambda_{\tau+1,\nu}Q_{\nu}(x)+\sum_{\tau=0}^{n}\lambda_{n,\tau}\lambda_{\tau+1,0}Q_{0}(x)\Big{]}$
$\displaystyle\
\,-{\xi_{n}}\mu_{n}\left(\gamma_{n+1}+\gamma_{n}\right)Q_{n-1}(x)-{\xi_{n}}\mu_{n-2}\gamma_{n}\gamma_{n-1}Q_{n-3}(x).$
(2.8)

From this, for $n\geqslant 1$, equating the coefficients of $Q_{n-1}$ leads to

$\displaystyle\xi_{n}\mu_{n+2}\lambda_{n+1,n-1}-\xi_{n+1}\mu_{n}\lambda_{n,n-2}+\xi_{n+1}\mu_{n-2}\gamma_{n}-\xi_{n}\mu_{n}\gamma_{n+1}=0.$
(2.9)

In particular, if we replace $n=1$ in the last equation we find

$\displaystyle\mu_{3}\lambda_{2,0}-\mu_{1}\left(\gamma_{2}+\gamma_{1}\right)=0.$

But, for $n=0$ in (2.1), we can write
$\gamma_{1}=(\mu_{3}\theta_{1}-2)\frac{\gamma_{2}}{\mu_{2}}$ giving

$\displaystyle\lambda_{2,0}=\frac{\mu_{1}}{\mu_{2}}\theta_{1}\gamma_{2}.$

We now proceed by induction and assume that the following formula holds

$\displaystyle\lambda_{n,n-2}=\frac{\mu_{n-1}}{\mu_{n}}\theta_{n-1}\gamma_{n},\
\ \mbox{ for}\ n\geqslant 2.$ (2.10)

Because of (2.1) and (2.10), we easily deduce from (2.9) that

$\displaystyle\lambda_{n+1,n-1}=\frac{\mu_{n}}{\mu_{n+1}}\theta_{n}\gamma_{n+1}\neq
0,\ n\geqslant 1.$

Similarly, for $n\geqslant 3$, equating the coefficients of $Q_{n-3}$ gives

$\displaystyle\xi_{n}\mu_{n+2}\lambda_{n+1,n-3}-\xi_{n+1}\mu_{n}\lambda_{n,n-4}+\left(2\mu_{n-1}\gamma_{n}-\xi_{n+1}\mu_{n}\lambda_{n,n-2}\right)\lambda_{n-1,n-3}-\xi_{n}\mu_{n-1}\gamma_{n}\gamma_{n-1}=0.$

After replacing $\lambda_{n,n-2}$ and $\lambda_{n-1,n-3}$ by their expressions
provided by (2.10), the latter becomes

$\displaystyle\xi_{n}\mu_{n+2}\lambda_{n+1,n-3}-\xi_{n+1}\mu_{n}\lambda_{n,n-4}=\mu_{n-2}\gamma_{n}\gamma_{n-1}\big{[}\xi_{n+1}\theta_{n-1}\theta_{n-2}-2\theta_{n-2}+\xi_{n}\big{]}.$
(2.11)

Since $\xi_{n+1}=\xi_{n-1}$ and so $\xi_{n}=\xi_{n-2}$, the Riccati equation
(2.2) readily implies the right-hand side of the latter equality to be zero
and therefore we find

$\displaystyle\xi_{n}\mu_{n+2}\lambda_{n+1,n-3}=\xi_{n+1}\mu_{n}\lambda_{n,n-4},\,n\geqslant
3.$ (2.12)

If we take $n=3$ in (2.12), we get $\lambda_{4,0}=0$, and recursively we
deduce that $\lambda_{n+1,n-3}=0$, for all $n\geqslant 3$, or, equivalently,
$\lambda_{n+4,n}=0$, for all $n\geqslant 0$.
We can now proceed analogously to show that $\lambda_{n+1,n+1-2k}=0$, for each
$k\geqslant 3$.
Equating the coefficients of $Q_{n+1-2k}$ for $k\geqslant 3$ in (2.8) leads to

$\displaystyle\xi_{n}\mu_{n+2}\lambda_{n+1,n+1-2k}+\left(2\mu_{n-1}\gamma_{n}-\xi_{n+1}\mu_{n}\lambda_{n,n-2}\right)\lambda_{n-1,n+1-2k}-\xi_{n+1}\mu_{n}\\!\\!\sum_{\tau=n-2k}^{n-6}\lambda_{n,\tau}\lambda_{\tau+1,n+1-2k}=0.$

Based on the identity (2.10), this may be also written as

$\displaystyle\xi_{n}\mu_{n+2}\lambda_{n+1,n+1-2k}+\mu_{n-1}\gamma_{n}\left(2-\xi_{n+1}\theta_{n-1}\right)\lambda_{n-1,n+1-2k}-\xi_{n+1}\mu_{n}\sum_{\tau=3}^{k}\lambda_{n,n-2\tau}\lambda_{n+1-2\tau,n+1-2k}=0.$

Suppose that $\lambda_{n,n-2j}=0$ for each $j=2,\ldots,k-1$, $n\geqslant 2j$.
This readily gives

$\displaystyle\xi_{n}\mu_{n+2}\lambda_{n+1,n+1-2k}=\xi_{n+1}\mu_{n}\lambda_{n,n-2k}.$
(2.13)

From the above, by decreasing $n$, it follows that

$\displaystyle\lambda_{n+1,n+1-2k}=\frac{\xi_{n+1}\mu_{2k+1}\mu_{2k}}{\xi_{2k}\mu_{n+2}\mu_{n+1}}\lambda_{2k,0}.$
(2.14)

In particular, for $n=2k$, we have

$\displaystyle\lambda_{2k+1,1}=\frac{\xi_{2k+1}\mu_{2k}}{\xi_{2k}\mu_{2k+2}}\lambda_{2k,0}.$
(2.15)

Finally, for $n\geqslant 6$, a simple comparison of the coefficients of
$Q_{0}$ in (2.8) leads to

$\displaystyle\xi_{n}\mu_{n+1}\lambda_{n+1,0}+2\mu_{n-1}\gamma_{n}\lambda_{n-1,0}-\xi_{n+1}\mu_{n}\sum_{\tau=0}^{n-1}\lambda_{n,\tau}\lambda_{\tau+1,0}=0.$
(2.16)

But, due to the symmetry property of $\\{Q_{n}\\}_{n\geqslant 0}$, we have
$\lambda_{2n+1,0}=0$ for $n\geqslant 0$. Accordingly, if we replace $n$ in
(2.16) by $2n+1$, this identity may be written as

$\displaystyle\xi_{2n+1}\mu_{2n+2}\lambda_{2n+2,0}+2\mu_{2n}\gamma_{2n+1}\lambda_{2n,0}-\xi_{2n+2}\mu_{2n+1}\lambda_{2n+1,1}\lambda_{2,0}-\xi_{2n+2}\mu_{2n+1}\sum_{\tau=1}^{n-2}\lambda_{2n+1,2\tau+1}\lambda_{2\tau+2,0}=0.$

Based on the identities (2.14)-(2.15), if we suppose that $\lambda_{2k,0}=0$,
$k=2,\ldots,n$, we deduce by induction that $\lambda_{2n+2,0}=0$, and then
obtain

$\displaystyle\lambda_{n+1,n+1-2k}=0,\quad\mbox{for any }\ k\geqslant 3.$

As a consequence of all the above results, we deduce that (2.6) writes

$\displaystyle xQ_{n}(x)=Q_{n+1}+\lambda_{n+1,n-1}Q_{n-1}(x),\,n\geqslant
0,\quad\mbox{with }\
\lambda_{n+1,n-1}=\frac{\mu_{n}}{\mu_{n+1}}\theta_{n}\gamma_{n+1}\neq 0.$

This means that $\\{Q_{n}\\}_{n\geqslant 0}$ is a symmetrical OPS which is the
desired conclusion.
Remark 1. Clearly, the identification of the polynomial sequences discussed
here depends on the solutions of the Riccati equation (2.2), which are given
by

$\displaystyle{\bf A.}\quad\theta_{n}$ $\displaystyle=1,\ \ n\geqslant
1,\quad\mbox{(the trivial solution)};$ $\displaystyle{\bf B.}\quad\theta_{n}$
$\displaystyle=\frac{n+\theta+1-\mu(-1)^{n}}{n+\theta+\mu(-1)^{n}},\ \
n\geqslant 1,\quad\mbox{with}\ \theta\neq-n\pm\mu(-1)^{n}\ \ \mbox{an
arbitrary parameter}.\hskip 142.26378pt$

In consequence, without imposing any additional conditions, we can then say
that there are exactly two families of symmetric Dunkl-classical OPS that are
solutions to our problem: the generalized Hermite polynomials (obtained in
Case A) and the generalized Gegenbauer polynomials (obtained in Case B).
It is worth mentioning that when $\mu=0$, the Dunkl operator coincides with
the usual differential operator, say $T_{0}=D$, and so the system (2.1)-(2.2)
is simplified to

$\displaystyle{\gamma_{n+2}}=\frac{{n+2}}{{n+1}}\frac{\big{(}n(\theta_{n}-1)+1\big{)}}{\big{(}{(n+3)}(\theta_{n+1}-1)+1\big{)}}{\gamma_{n+1}},\
n\geqslant 0,$ $\displaystyle\theta_{n+1}+{\theta_{n}^{-1}}=2,\ n\geqslant 1.$

This system was already solved providing the two families of symmetric
classical orthogonal polynomials: Hermite polynomials and Gegenbauer
polynomials (see [2] for more details).
Before discussing the corresponding results when $\mu\neq 0$ with the
identification of the obtained OPS as announced above, we first give the next
theorem which is an important consequence of Theorem 2.1. Note that a partial
result of this theorem was presented in [3, Lemma 2.1], however, our proof is
quite different from that of this paper. Here, we also prove the reciprocal
condition found in this work.

###### Theorem 2.2

Let $\\{P_{n}\\}_{n\geqslant 0}$ be a symmetric MOPS satisfying the recurrence
relation (1.11a)-(1.11b). Then $\\{P_{n}\\}_{n\geqslant 0}$ is
$T_{\mu}$-classical if and only if each polynomial $P_{n}$ satisfies the
differential-difference equation

$\displaystyle\left(a_{n}x^{2}-b_{n}\right)T^{2}_{\mu}P_{n}-c_{n}xT_{\mu}P_{n}-d_{n}P_{n}=0,$
(2.17)

where

$\displaystyle a_{n}$ $\displaystyle=\theta_{n-1}-1,$ (2.18a) $\displaystyle
b_{n}$
$\displaystyle={\left(\mu_{n+1}\theta_{n-1}-\mu_{n}\right)\left(\mu_{n-1}\theta_{n-1}-\mu_{n-2}\right)}\frac{\gamma_{n}}{\xi_{n}\mu_{n}},$
(2.18b) $\displaystyle c_{n}$ $\displaystyle=\mu_{n-2}\theta_{n-1}-\mu_{n-1},$
(2.18c) $\displaystyle d_{n}$ $\displaystyle=\xi_{n}\mu_{n}\theta_{n-1},$
(2.18d)

with $\theta_{n}$ a solution to the Riccati equation (2.2). It is understood
that $\theta_{-1}=0$.

Before proving Theorem 2.2, it must be noted that the expression defining
$b_{n}$ is initially given by

$b_{n}={\left(\mu_{n+1}\theta_{n-1}-\mu_{n}\right)\left(\mu_{n+2}\theta_{n}-\mu_{n+1}\right)}\frac{\gamma_{n+1}}{\xi_{n+1}\mu_{n+1}}.$
(2.19)

Indeed, after the substitution of the coefficient $\gamma_{n+1}$ by an
expression in terms of $\gamma_{n}$ provided in (2.1) we see at once that
(2.19) is an equivalent form of (2.18b).
Proof The necessary condition. If we replace $n$ in (1.13a) by $n-1$, taking
into account (2.3), we get

$\displaystyle\mu_{n+1}Q_{n+1}(x)=\mu_{n+1}xQ_{n}(x)-\mu_{n}\theta_{n}\gamma_{n+1}Q_{n-1}(x),\
n\geqslant 0,$ (2.20)

On combining this with (1.16) yields

$\displaystyle\xi_{n+1}\mu_{n+1}P_{n+1}(x)+\mu_{n}\left(\mu_{n+2}\theta_{n}-\mu_{n+1}\right)\gamma_{n+1}Q_{n-1}(x)-\xi_{n+1}\mu_{n+1}xQ_{n}(x)=0,\
n\geqslant 0.$ (2.21)

Multiply (2.21) by $\theta_{n}$ and use (2.20), then shift $n\rightarrow n-1$,
we obtain

$\displaystyle\xi_{n}\theta_{n-1}P_{n}(x)+\mu_{n}\left(\theta_{n-1}-1\right)xQ_{n-1}(x)-\left(\mu_{n+1}\theta_{n-1}-\mu_{n}\right)Q_{n}(x)=0,\
n\geqslant 0.$ (2.22)

If we multiply now (2.21) by $\left(\mu_{n+1}\theta_{n-1}-\mu_{n}\right)$ and
(2.22) by $\xi_{n+1}\mu_{n+1}x$ and add the resulting identities, it follows
that

$\displaystyle\left(\mu_{n+1}\theta_{n-1}-\mu_{n}\right)P_{n+1}(x)$
$\displaystyle=\xi_{n}\theta_{n-1}xP_{n}(x)+\left(\theta_{n-1}-1\right)x^{2}T_{\mu}P_{n}(x)$
$\displaystyle-\left(\mu_{n+2}\theta_{n}-\mu_{n+1}\right)\left(\mu_{n+1}\theta_{n-1}-\mu_{n}\right)\frac{\gamma_{n+1}}{\xi_{n+1}\mu_{n+1}}T_{\mu}P_{n}(x).$

The action of $T_{\mu}$ on both sides of the latter gives us

$\displaystyle\left(\mu_{n+1}\theta_{n-1}-\mu_{n}\right)T_{\mu}P_{n+1}(x)$
$\displaystyle=\left(a_{n}x^{2}-b_{n}\right)T^{2}_{\mu}P_{n}+\left((\xi_{n}+2)\theta_{n-1}-2\right)xT_{\mu}P_{n}+\xi_{n}^{2}\theta_{n-1}P_{n}(x).$
(2.23)

On the other hand, we see that (2.22) writes also in the form

$\displaystyle\left(\mu_{n+1}\theta_{n-1}-\mu_{n}\right)T_{\mu}P_{n+1}(x)=\mu_{n+1}\xi_{n}\theta_{n-1}P_{n}(x)+\mu_{n+1}\left(\theta_{n-1}-1\right)xT_{\mu}P_{n}(x).$
(2.24)

Finally, equating the right hand sides of both (2.23) and (2.24), the
differential-difference equation (2.17) follows immediately with the
coefficients $a_{n},\ldots,d_{n}$ are defined by (2.18a)-(2.18d).
Conversely, suppose that the MOPS $\\{P_{n}\\}_{n\geqslant 0}$ fulfils (2.17)
with the definitions (2.18a)-(2.18d). Our objective is to establish the
orthogonality of the sequence $\\{Q_{n}\\}_{n\geqslant 0}$.
Since the polynomials $P_{n}$ are monic and symmetric, if we write
$P_{n}(x)=x^{n}+\alpha_{n}x^{n-2}+\cdots$, for $n\geqslant 2$, and insert this
in (1.11a), make the change $n\rightarrow n-1$, a simple identification gives
rise to

$\displaystyle\gamma_{n}=\alpha_{n}-\alpha_{n+1},\ n\geqslant 1.$ (2.25)

Besides, substitute the above expansion of $P_{n}(x)$ in the differential-
difference equation (2.17), we easily check that the coefficients of $x^{n}$
and $x^{n-2}$ provide the following equalities

$\displaystyle d_{n}=\mu_{n}\mu_{n-1}a_{n}-\mu_{n}c_{n},\ n\geqslant 0,$
(2.26)

and

$\displaystyle\big{(}\mu_{n-2}\mu_{n-3}a_{n}-\mu_{n-2}c_{n}-d_{n}\big{)}\alpha_{n}=\mu_{n}\mu_{n-1}b_{n},\
n\geqslant 0.$ (2.27)

On combining (2.26) with (2.27), taking into consideration (2.18a)-(2.18d), we
see that

$\displaystyle\alpha_{n}=-\frac{\mu_{n}\mu_{n-1}b_{n}}{2\big{(}\mu_{n-1}\theta_{n-1}-\mu_{n-2}\big{)}},\
n\geqslant 0.$ (2.28)

Therefore (2.25) is readily written as

$\displaystyle\gamma_{n}=\frac{\mu_{n+1}\mu_{n}b_{n+1}}{2\big{(}\mu_{n}\theta_{n}-\mu_{n-1}\big{)}}-\frac{\mu_{n}\mu_{n-1}b_{n}}{2\big{(}\mu_{n-1}\theta_{n-1}-\mu_{n-2}\big{)}},\
n\geqslant 1.$ (2.29)

Now, we substitute in the latter $b_{n+1}$ (resp. $b_{n}$) by its respective
expression provided from the identity (2.18b) (resp.(2.19)) obtaining

$\displaystyle
2\xi_{n+1}\mu_{n+1}\big{(}\mu_{n-1}\theta_{n-1}-\mu_{n-2}\big{)}\gamma_{n}=$
$\displaystyle\left[\mu_{n+1}\big{(}\mu_{n-1}\theta_{n-1}-\mu_{n-2}\big{)}-\mu_{n-1}\big{(}\mu_{n+1}\theta_{n-1}-\mu_{n}\big{)}\right]\times$
$\displaystyle\
\mu_{n}{\big{(}\mu_{n+2}\theta_{n}-\mu_{n+1}\big{)}}\gamma_{n+1},\ n\geqslant
1.$ (2.30)

An easy computation shows that the expression between square brackets
simplifies as

$\mu_{n+1}\big{(}\mu_{n-1}\theta_{n-1}-\mu_{n-2}\big{)}-\mu_{n-1}\big{(}\mu_{n+1}\theta_{n-1}-\mu_{n}\big{)}=2\xi_{n}.$

Substituting this into (2.30) and then replacing $n$ by $n+1$ we conclude that

$\displaystyle{\gamma_{n+2}}=\frac{\xi_{n+2}}{\xi_{n+1}}\frac{\mu_{n+2}}{\mu_{n+1}}\frac{\big{(}\mu_{n}\theta_{n}-\mu_{n-1}\big{)}}{\big{(}\mu_{n+3}\theta_{n+1}-\mu_{n+2}\big{)}}{\gamma_{n+1}},\
n\geqslant 0,$

and so we have found the equality (2.1). In consequence, use of Theorem 2.1
ensures the orthogonality of the sequence $\\{Q_{n}(x)\\}$, which means that
$\\{P_{n}(x)\\}$ is $T_{\mu}$-classical, and the proof is complete.
Remark 2. As we observed in Remark 1, for $\mu=0$, we again meet each of the
families of Hermite polynomials and Gegenbauer polynomials. This will be more
specified in the next section when applying Theorem 2.2. Equation (2.17) is
then reduced to the ordinary differential equation of type

$\displaystyle a(x)y^{\prime\prime}+b(x)y^{\prime}-\lambda_{n}y=0,\ \
y=P_{n}(x),$ (2.31)

where $a(x)$ is a $0$-degree polynomial or a $2$-degree polynomial (depending
on the cases A or B), $b(x)$ is a polynomial with degree $1$ and $\lambda_{n}$
are scalars.

##  3 Applications

We illustrate Theorems 2.1 and 2.2 by considering separately the two cases A
and B as referred to above. We confirm that the only symmetric Dunkl-classical
OPS are the generalized Hermite polynomials and the generalized Gegenbauer
polynomials, as one may have expected. As we have seen, this result was
obtained directly from the solutions of Equation (2.2) without imposing any
further restrictions.
In addition to illustrating how our theorems provide the above results, we
also remind some of the known properties satisfied by the resulting
polynomials. We use, for the most part, the notations of [7].

Case A. Application of Theorem 2.1 when $\theta_{n}=1$ shows that Equality
(2.1) is reduced to

$\displaystyle\gamma_{n+2}=\frac{\mu_{n+2}}{\mu_{n+1}}\gamma_{n+1},\
n\geqslant 0,$

from which, taking into account (2.3), we immediately deduce that
$\tilde{\gamma}_{n+1}=\gamma_{n+1}=\frac{\gamma_{1}}{\mu_{1}}{\mu_{n+1}},\
n\geqslant 0$.
Letting $\delta_{n}=\mu(1-(-1)^{n})$ and choosing
$\gamma_{1}=\frac{1}{2}\mu_{1}$, it follows that

$\displaystyle\tilde{\gamma}_{n+1}=\gamma_{n+1}=\frac{1}{2}\big{(}n+1+\delta_{n+1}\big{)},\
n\geqslant 0.$ (3.1)

From this we see that the regularity conditions still hold provided
$\mu\neq-n-\frac{1}{2},n=0,1,2,\ldots$. Moreover, it follows from (3.1) that
$Q_{n}=P_{n},\,n\geqslant 0$, which reads as
$T_{\mu}P_{n+1}=\mu_{n+1}P_{n},\,n\geqslant 0$. Whence the OPS
$\\{P_{n}\\}_{n\geqslant 0}$ belongs to the class of Appell sequences relative
to $T_{\mu}$. It said to be a Dunkl-Appell sequence.
We thus encounter the generalized Hermite polynomials
$\\{H^{\mu}_{n}\\}_{n\geqslant 0}$ which have been studied extensively by
Chihara in his Ph.D. thesis [6]. The same author was also established in [7]
that these polynomials are related to the generalized Laguerre polynomials
$L_{n}^{\alpha}(x),\,n=0,1,\ldots$, via

$\displaystyle H^{\mu}_{2n}(x)$
$\displaystyle=(-1)^{n}2^{2n}n!L_{n}^{\alpha}(x^{2}),\ n\geqslant 0,$
$\displaystyle H^{\mu}_{2n+1}(x)$
$\displaystyle=(-1)^{n}2^{2n+1}n!xL_{n}^{\alpha+1}(x^{2}),\ n\geqslant
0,\quad\alpha=\mu-\frac{1}{2}.$

In particular, for $\mu=0$, we have $H^{0}_{n}(x)={H}_{n}(x),n\geqslant 0$,
the classical Hermite polynomials.
The normalization coefficient is chosen so that the leading coefficient of
$H^{\mu}_{n}(x)$ becomes $2^{n}$ and so
$H^{\mu}_{n}(x)=2^{n}\hat{H}^{\mu}_{n}(x)$, which means that the sequence
$\\{P_{n}\\}_{n\geqslant 0}$ coincides with
$\\{\hat{H}^{\mu}_{n}\\}_{n\geqslant 0}$.
Chihara pointed out that these polynomials were first introduced by Szegö in
[24, p.380] who gave the associated orthogonalizing weight
$w(x)=|x|^{2\mu}e^{-x^{2}}$, $\mu>-\frac{1}{2}$, supported on
$(-\infty,+\infty)$ and found that each polynomial satisfies two second order
differential equations. For instance, the following

$\displaystyle
xy^{\prime\prime}+2(\mu-x^{2})y^{\prime}+(2nx-\delta_{n}x^{-1})y=0.$ (3.2)

Many other formulas may be found in [7, pp.156-158]. On the other hand, Maroni
[18] obtained that the generalized Hermite polynomials satisfy the structure
relation

$\displaystyle
xP^{\prime}_{n+1}(x)=-\delta_{n+1}P_{n+1}(x)+2\gamma_{n+1}xP_{n}(x),\quad
P_{n}(x)=\hat{H}^{\mu}_{n}(x),$

which can also be turned into

$\displaystyle
xP^{\prime}_{n+1}(x)=(n+1)P_{n+1}(x)+\frac{1}{2}\left(n^{2}+(2\mu+1)n+\delta_{n}\right)\\!P_{n-1}(x).$

For $\mu=0$, one sees that $Q_{n}(x)=P_{n}(x)=(n+1)^{-1}P^{\prime}_{n+1}(x)$.
In consequence, the latter becomes

$\displaystyle xQ_{n}(x)=Q_{n+1}(x)+\frac{1}{2}nQ_{n-1}(x),\ n\geqslant 0,$

which is clearly the recurrence relation satisfied by the derivatives of the
classical Hermite polynomials.

Among the various works concerning applications of generalized Hermite
polynomials, we quote, for instance, [21, 22, 23] (see also the references
given there).

Now let $\mathscr{H}_{\mu}$ be the linear functional associated to the
polynomials ${H}^{\mu}_{n},n\geqslant 0$. The regularity of this functional is
guaranteed provided $\mu\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\ldots$,
and it is positive definite only when $\mu>-\frac{1}{2}$.
By considering some special cases of semi-classical forms, it was observed in
[18] that $\mathscr{H}_{\mu}$ is actually a semi-classical functional of class
one satisfying the distributional differential equation (1.9), where
$\phi(x)=x$ and $\psi(x)=2x^{2}-(2\mu+1)$. This clearly specifies that
$s=\max\left(\deg\phi-2,\deg\psi-1\right)=1$.
As above-mentioned, for $\mu>-\frac{1}{2}$, this functional is represented as

$\displaystyle\big{<}\mathscr{H}_{\mu},f\big{>}=\int_{-\infty}^{+\infty}f(x)|x|^{2\mu}e^{-x^{2}}dx,\
\ \forall f\in\mathscr{P},$

so that its moments are given by

$\displaystyle(\mathscr{H}_{\mu})_{2n}={\Gamma\big{(}n+\mu+{1}/{2}\big{)}},\
n\geqslant 0,$ (3.3a) $\displaystyle(\mathscr{H}_{\mu})_{2n+1}=0,\ n\geqslant
0.$ (3.3b)

When $\mu<-\frac{1}{2}$, however, finding orthogonalizing weight function is
more difficult. Morton and Krall [20] attacked this problem from the point of
view of distribution theory. In fact, based on the canonical regularization
distributional technique introduced in [13, Ch.1, Sec.3], the authors had
developed a procedure for constructing distributional weight functions
supported on a succession of disjointed strips.
Applying this method, Krall showed in [16] that an integral representation of
the functional $\mathscr{H}_{\mu}$ can be obtained, in any strip of the form
$-j-\frac{1}{2}<\mu<-j+\frac{1}{2}$, where $j$ is a fixed positive integer, in
the form

$\displaystyle\big{<}\mathscr{H}_{\mu},f\big{>}=\int_{0}^{+\infty}\\!\\!|x|^{2\mu}\Big{\\{}e^{-x^{2}}[f(x)+f(-x)]-\\!\\!\sum_{k=0}^{2j-2}\bigl{(}e^{-x^{2}}[f(x)+f(-x)]\bigr{)}^{(k)}(0)\frac{x^{k}}{k!}\Big{\\}}dx,$
(3.4)

where $f$ is a sufficiently differentiable function and $g^{(k)}$ stands for
the $k^{th}$ derivative of the function $g$. Note that in this case there is a
double regularization of $|x|^{2\mu}$ at $x=0$ following Gelfand and Shilov
[13].
The $2j$-fold integration by parts in (3.4) shows that

$\displaystyle\big{<}\mathscr{H}_{\mu},f\big{>}=\frac{\Gamma(2\mu+1)}{\Gamma(2\mu+2j+1)}\int_{-\infty}^{+\infty}\\!|x|^{2\mu+2j}\big{(}e^{-x^{2}}f(x)\big{)}^{(2j)}dx,\
\ j=1,2,\ldots.$ (3.5)

A third form of the distributional weight function involving the classical
Hermite polynomials has also been given in [16]. This representation is fully
equivalent to both (3.4) and (3.5) when $f$ is a polynomial. In doing so, the
author had taken as the only starting point the moments (3.3a)-(3.3b), under
the regularity conditions imposed on $\mu$. For more details about this
procedure we refer the reader to [20, 16].

We continue in this fashion with Theorem 2.2 to obtain that (2.18a)-(2.18b)
simplify to

$a_{n}=0,\,b_{n}=\xi_{n}/2,\,c_{n}=-\xi_{n}\ \,\mbox{and}\
\,d_{n}=\xi_{n}\mu_{n}.$

On substituting these into (2.17) we obtain

$\displaystyle T^{2}_{\mu}P_{n}-2xT_{\mu}P_{n}+\lambda_{n}P_{n}=0,$ (3.6)

where $\lambda_{n}=2\mu_{n}=2\big{(}n+\delta_{n}\big{)},\ n\geqslant 0,$ or,
equivalently,

$\displaystyle\lambda_{2n}=4n,\ n\geqslant 0,\ \ \mbox{and}\ \
\lambda_{2n+1}=2\big{(}2n+2\mu+1\big{)},\ n\geqslant 0.$

For $\mu=0$, we easily check that (3.6), (3.2) are identical and coincide with
(2.31), where $a(x)=1$, $b(x)=-2x$ and $\lambda_{n}=-2n$. It follows that the
equation outlined here is the second-order differential equation satisfied by
the ordinary Hermite polynomials $H_{n},\,n=0,1,\ldots$.

Case B. Similarly to the previous case, on account of the value of
$\theta_{n}$, we can write (2.1) in the form

$\displaystyle{\gamma_{n+2}}=\frac{\big{(}2n+\theta+\mu\big{)}\big{(}n+1+\theta-\mu(-1)^{n}\big{)}}{\big{(}2(n+2)+\theta+\mu\big{)}\big{(}n+\theta+\mu(-1)^{n}\big{)}}\frac{\mu_{n+2}}{\mu_{n+1}}{\gamma_{n+1}},\
n\geqslant 0.$ (3.7)

Multiply, in (3.7), the numerator and denominator of the right-hand side by
$2(n+1)+\theta+\mu$ to readily deduce by telescopy that

$\displaystyle\gamma_{n+1}=\big{(}\theta+\mu+2\big{)}\frac{\gamma_{1}}{\mu_{1}}\frac{\big{(}n+\theta+\mu(-1)^{n}\big{)}\big{(}n+1+\delta_{n+1}\big{)}}{\big{(}2n+2+\theta+\mu\big{)}\big{(}2n+\theta+\mu\big{)}},\
n\geqslant 0.$

For two fixed parameters $\alpha$ and $\beta$, if we let
$\gamma_{1}=\frac{\beta+1}{\alpha+\beta+2}$ and set
$\,\theta+\mu=2\big{(}\alpha+\beta+1\big{)},\ \theta-\mu=2\alpha+1$, we get
$2\mu=2\beta+1$ and $\mu_{1}=2(\beta+1)$. Thus, the above expression may be
rewritten in the form [1]:

$\displaystyle\gamma_{n+1}=\frac{\big{(}n+1+2\alpha+\delta_{n+1}\big{)}\big{(}n+1+\delta_{n+1}\big{)}}{4\big{(}n+2+\alpha+\beta\big{)}\big{(}n+1+\alpha+\beta\big{)}},\
n\geqslant 0.$ (3.8)

With the above reparameterization, we must take into account that the validity
of (3.8) presumes the parameters $\alpha$ and $\beta$ to take values in
$\mathbb{C}$ within the range of regularity.
The polynomials arising in this case are clearly the generalized Gegenbauer
ones which we denote here by $\\{{G}^{(\alpha,\beta)}_{n}\\}_{n\geqslant 0}$.
These polynomials were first investigated by Chihara [7] who obtained their
main properties by employing a direct relation between them and the Jacobi OP
$P_{n}^{(\alpha,\beta)},\ n\geqslant 0$:

$\displaystyle{G}^{(\alpha,\beta)}_{2n}(x)$
$\displaystyle=P_{n}^{(\alpha,\beta)}(2x^{2}-1),\ n\geqslant 0,$
$\displaystyle{G}^{(\alpha,\beta)}_{2n+1}(x)$
$\displaystyle=xP_{n}^{(\alpha,\beta+1)}(2x^{2}-1),\ n\geqslant 0.$

With the notations $\hat{P}_{n}^{(\alpha,\beta)}$ and
$\hat{G}_{n}^{(\alpha,\beta)}$ for the monic Jacobi polynomials and
generalized Gegenbauer polynomials, respectively, we have

$\displaystyle\hat{G}^{(\alpha,\beta)}_{2n}(x)={2^{-n}}\hat{P}_{n}^{(\alpha,\beta)}(2x^{2}-1)\quad\mbox{and}\
\
\hat{G}^{(\alpha,\beta)}_{2n+1}(x)={2^{-n}}\hat{P}_{n}^{(\alpha,\beta+1)}(2x^{2}-1).$
(3.9)

So, the resulting polynomials $\\{P_{n}\\}_{n\geqslant 0}$ obtained in this
second case coincide with $\\{\hat{G}^{(\alpha,\beta)}_{n}\\}_{n\geqslant 0}$,
and are therefore orthogonal with respect to the weight function
$w(x)=|x|^{2\mu}(1-x^{2})^{\alpha}$ on $(-1,+1)$.
Furthermore, combining (2.3) with (3.8), we see at once that the coefficients
$\tilde{\gamma}_{n},\,n\geqslant 1$, defining the sequence
$\\{Q_{n}\\}_{n\geqslant 0}$ are given by

$\displaystyle\tilde{\gamma}_{n+1}=\frac{\big{(}n+3+2\alpha+\delta_{n+1}\big{)}\big{(}n+1+\delta_{n+1}\big{)}}{4\big{(}n+3+\alpha+\beta\big{)}\big{(}n+2+\alpha+\beta\big{)}},\
n\geqslant 0.$ (3.10)

Now, a simple comparison between (3.8) and (3.10) shows that

$\displaystyle\tilde{\gamma}_{n+1}(\alpha,\beta)=\gamma_{n+1}(\alpha+1,\beta),\
n\geqslant 0,$ (3.11)

where $\gamma_{n+1}:=\gamma_{n+1}(\alpha,\beta)$ and
$\tilde{\gamma}_{n+1}:=\tilde{\gamma}_{n+1}(\alpha,\beta)$. So, with the
notations $P_{n}(x):=P_{n}(x;\alpha,\beta),\,n\geqslant 0$, and
$Q_{n}(x):=Q_{n}(x;\alpha,\beta),\,n\geqslant 0$, we have
$Q_{n}(x;\alpha,\beta)=P_{n}(x;\alpha+1,\beta),\,n\geqslant 0$, which means
that

$\displaystyle
T_{\mu}P_{n+1}(x;\alpha,\beta)=\mu_{n+1}P_{n}(x;\alpha+1,\beta),\,n\geqslant
0.$ (3.12)

A number of works in different contexts have been devoted to the generalized
Gegenbauer polynomials (see for instance [15, 5, 8, 1]) highlighting several
properties of these polynomials. For example, it was obtained in [1] that
these polynomials satisfy the structure relation

$\displaystyle
x(x^{2}-1)P^{\prime}_{n+1}(x)=\left((n+1)x^{2}+\delta_{n+1}\right)P_{n+1}(x)-2\gamma_{n+1}(n+\alpha+\beta+2)xP_{n}(x),$

where the coefficients $\gamma_{n},\,n\geqslant 1$, are given by (3.8). An
equivalent formulation of this relation is

$\displaystyle x(x^{2}-1)P^{\prime}_{n+1}(x)$
$\displaystyle=(n+1)P_{n+3}(x)-\left[(n+1)\gamma_{n+2}+\big{(}3(n+1)+2(\alpha+\beta+1)\big{)}\gamma_{n+1}-\delta_{n+1}\right]P_{n+1}(x)$
$\displaystyle\,+\gamma_{n+1}\gamma_{n}\big{(}3(n+1)+2(\alpha+\beta+1)\big{)}P_{n-1}(x).$

The same author also showed that each polynomial satisfies the second order
differential equation

$\displaystyle
x^{2}(x^{2}-1)y^{\prime\prime}+x\left[(2\alpha+2\beta+3)x^{2}-2\beta-1\right]y^{\prime}-\left[n(n+2\alpha+2\beta+2)x^{2}-\delta_{n}\right]y=0.$
(3.13)

If moreover $\mathscr{G}_{\alpha,\beta}$ denotes the Gegenbauer linear
functional associated to the generalized Gegenbauer polynomials, he found that
$\mathscr{G}_{\alpha,\beta}$ satisfies (1.9), where $\phi(x)=x(x^{2}-1)$ and
$\psi(x)=-\left(2(\alpha+\beta+2)x^{2}+\beta+1\right)$, with $\alpha,\beta>-1$
and $\beta\neq-\frac{1}{2}$. From this, we see that
$\mathscr{G}_{\alpha,\beta}$ is also a semi-classical functional of class
$s=1$. The linear functional $\mathscr{G}_{\alpha,\beta}$ admits then the
integral representation

$\displaystyle\big{<}\mathscr{G}_{\alpha,\beta},f\big{>}=\int_{-1}^{+1}\\!f(x)|x|^{2\mu}(1-x^{2})^{\alpha}dx,\
\forall f\in\mathscr{P},$

and it is positive definite when $\alpha,\beta>-1$. Its moments are given by

$\displaystyle(\mathscr{G}_{\alpha,\beta})_{2n}=\frac{\Gamma\big{(}\alpha+1\big{)}\Gamma\big{(}n+\beta+1\big{)}}{\Gamma\big{(}n+\alpha+\beta+2\big{)}},\
n\geqslant 0,$ (3.14a) $\displaystyle(\mathscr{G}_{\alpha,\beta})_{2n+1}=0,\
n\geqslant 0,$ (3.14b)

which are valid for $\alpha+\beta\neq-n,n=1,2,\ldots$,
$\beta\neq-n,n=1,2,\ldots$. In the above, the case $\beta\neq-1/2$ has to be
excluded since this involves the ultraspherical or Gegenbauer polynomials
${G}_{n}^{\nu}$, with $\nu=\alpha+1/2$.
Now, how can finding an orthogonalizing weight function for the polynomials
${G}_{n}^{(\alpha,\beta)}$ when $\alpha,\beta\ngtr-1$ on an interval of the
real line. Here as well as in the previous case, the existence of such a
weight is in fact guaranteed by Boas’s result [4], and it may be constructed
in any region in which $\alpha$, $\beta$ belong to the strips $-i-1<\alpha<-i$
and $-j-1<\beta<-j$, respectively, where $i$, $j$ are two positive integers.
To do so, one can proceed in three independent parts to regularize
$w(x)=|x|^{2\mu}(1-x^{2})^{\alpha}$ at $x=0$ and $x=\pm 1$.
Notice that the process requires a tedious computation in this case that we do
not present here.
However, to have an idea about what we can expect when $\alpha,\beta\ngtr-1$,
we refer the reader to [20] where a detailed discussion concerning this
technique was done, including the Jacobi case.
We now return to apply Theorem 2.2. First, a trivial verification shows that

$a_{n}=b_{n}=\frac{\xi_{n}}{n+\theta-1-\mu(-1)^{n}},\,c_{n}=-\frac{2(\alpha+1)\xi_{n}}{n+\theta-1-\mu(-1)^{n}}\
\ \mbox{and}\
\,d_{n}=\xi_{n}\mu_{n}\frac{n+\delta_{n+1}+2\alpha+1}{n+\theta-1-\mu(-1)^{n}}.$

Next, when these are substituted in (2.17) we obtain after simplification that

$\displaystyle(1-x^{2})T^{2}_{\mu}P_{n}-2(\alpha+1)xT_{\mu}P_{n}+\lambda_{n}P_{n}=0,$
(3.15)

where $\lambda_{n}=\mu_{n}\big{(}n+\delta_{n+1}+2\alpha+1\big{)},\ n\geqslant
0,$ which also writes

$\displaystyle\lambda_{2n}=4n\big{(}n+\alpha+\beta+1\big{)},\ n\geqslant 0,\ \
\mbox{and}\ \
\lambda_{2n+1}=4\big{(}n+\alpha+1\big{)}\big{(}n+\beta+1\big{)},\ n\geqslant
0.$

In this case, when $\mu=0$, we easily check that (3.15), (3.13) are fully
equivalent and deduce that they reduce to (2.31) with $a(x)=1-x^{2}$,
$b(x)=-(2\nu+1)x$ and $\lambda_{n}=n(n+2\nu)$, where $\nu$ stands for
$\alpha+1/2$.
Clearly, when $\mu$ assumes the value $0$, the three equations coincide and
provide the well-known differential equation satisfied by the Gegenbauer
polynomials ${G}_{n}^{\nu},\,n=0,1,\ldots.$

Conclusion.
We studied the symmetric Dunkl-classical orthogonal polynomials in another
way. This alternative approach for obtaining these polynomials has the
advantage of being self-contained for determining the recurrence coefficients
explicitly. A new characterization connecting the recurrence coefficients with
the Dunkl-classical character is established in Theorem 2.1, while the second
theorem (Theorem 2.2) provided another characterization of these polynomials
by means of a second order differential-difference equation. Direct
application of these two theorems readily showed that the only symmetric
Dunkl-classical orthogonal polynomials are the generalized Hermite and
generalized Gegenbauer polynomials. These results are of course expected and
consistent with those of [3].

## References

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# Stability of two-fluid galactic disc under the influence of an external
tidal field.

K. Aditya
Indian Institute of Science Education and Research, Tirupati 517507, India
E-mail<EMAIL_ADDRESS>

###### Abstract

We consider the dynamics of rotationally supported thin galactic disc composed
of stars and gas under the influence of external tidal field and derive the
coupled differential equations governing the evolution of instabilities.
Further linearising the governing equation a modified dispersion relation and
stability criterion for appraising the stability of the two fluid galactic
disc under the influence of external tidal field is obtained. Possible
applications and method for the same are discussed.

_K_ eywords :

hydrodynamics-instabilities :galaxies:kinematics and dynamics-
galaxies:structure-galaxies:star formation

##  1 Introduction

Local stability of the galactic disc is a subtle balance between the mass
contained in the fluid packet, the random velocity dispersion and the
differential rotation. The competetion between the stabilising agents i.e
differential rotation and the random velocity dispersion and the destabilising
agent i.e the mass content is classically quantified by the stability
criterion proposed by Toomre (1964).

$Q=\frac{\kappa\sigma}{\pi G\Sigma}$ (1)

Where $\rm\kappa$ is the epicyclic frequency, $\rm\sigma$ is the radial
velocity dispersion, and $\rm\Sigma$ the surface mass density. Value of $\rm
Q\geq 1$ quantifies the stability of galactic disc against axis symmetric
perturbations. The the above criterion appraises the stability a fluid disc
consisting of either gas or stars at a time, but a real galactic disc consists
of stars and gas which is also gravitationally important, more so in case of
gas rich low surface brightness galaxies (LSBs) De Blok et al. (1996). The
dispersion relation pertaining to the stability of gravitationally coupled
two-fluid disc (stars+gas) has been extensively studied by Jog and Solomon
(1984) and criterion for appraising the stability of two-fluid disc has been
derived and studied by Elmegreen (1995), Jog (1996), and the same for multi-
component disc has been studied by Rafikov (2001). Stability criterion for
multi-component galactic disc with finite thickness has been studied by Romeo
and Falstad (2013), which is useful when the galactic disc consists of say
mutiple stellar disc (thin disc+thick disc) and HI disc. The role of external
tidal field on the stability of gravitating, infinite,homogenous systems was
investigated by Jog (2013) and modified Toomre stability criterion due to
external tidal field also derived by Jog (2014). In this work we consider the
simple case of an gravitationally interacting axis-symmetric isothermal fluid
disc of stars and gas supported by differential rotation and the random
velocity dispersion, under the influence of an external tidal field. We will
derive the basic differential equations governing the growth of density
perturbation and further obtain stability criterion for the two-component
fluid disc in presence of external tidal field. The paper is organised as
follows: in section 2 we will formulate the basic equations and derive the
governing differential equation, dispersion relation modified due to external
tidal fields, and an effective stability parameter, in section 3 we will
discuss possible applications and conclude in section 4.

##  2 Formulation and derivation of basic equations

The stars and gas in the galactic disc togather constitute isothermal fluids
interacting with each other gravitationally. The fluid disc is supported by
random presure and rotation. The problem is completely described by
understanding the flow of perturbed fluid components constituting a thin
cylindrical disc, i.e lying in z=0 plane. The dynamics of the perturbed fluid
and the conservation of mass density are dictated by the linearised force
equation (eq 2,3) and the linearised continuity equation (eq 4) for the two
-fluid system in cylindrical coordinates. The presence of an external tidal
field is included by adding an extra force
$\frac{-\partial\phi_{ext}}{\partial r}$ in the linearised force equation. The
dynamics of the fluid in force equation is governed by the self gravity of the
two-fluid disc, i.e each component of the perturbed fluid say ’gas’ moves
under the combined potential of stars + gas, thus the right hand side of the
linear force equation is supplimented with the linearised Poisson equation (eq
5) describing the combined mass densities of the two-fluid system and an
isothermal equation of state $\rm P_{i}=\Sigma_{i}c_{i}^{2}$ is used to
describe the presure of the perturbed fluids, where $\Sigma_{i}$ denotes the
surface density and $\rm c_{i}$ denotes the isothermal speed of sound of the
of the $i^{th}$ fluid component respectively. Throughout this work
$\rm{}^{\prime}i^{\prime}$ will index stars $(s)$ and gas $(g)$.

The linearised force equations under influence of an external tidal force are;

$\frac{\partial u_{i}}{\partial t}+\Omega_{0}\frac{\partial
u_{i}}{\partial\theta}-2\Omega_{0}v_{i}+\frac{c_{i}^{2}}{\Sigma_{0i}}\frac{\partial\Sigma_{i}}{\partial
R}+\frac{\partial(\phi_{g}+\phi_{s})}{\partial
R}+\frac{\Sigma_{i}}{\Sigma_{0i}}\frac{\partial\phi_{ext}}{\partial R}=0$ (2)

$\frac{\partial v_{i}}{\partial t}-2Bu_{i}+\Omega_{0}\frac{\partial
v_{i}}{\partial\theta}+\frac{1}{R}\frac{\partial(\Sigma_{s}+\Sigma_{g})}{\partial\theta}=0$
(3)

Linearised continuity equation reads;

$\frac{\partial\Sigma_{i}}{\partial
t}+\Omega_{0}\frac{\partial\Sigma_{i}}{\partial\theta}+\frac{\partial(\Sigma_{0i}u_{i})}{\partial
R}+\frac{u_{i}\Sigma_{0i}}{R}+\frac{\Sigma_{0i}}{R}\frac{\partial
v_{i}}{\partial\theta}=0$ (4)

Linearised Poisson equation reads;

$\frac{1}{R}\frac{\partial}{\partial
R}R\frac{\partial(\phi_{s}+\phi_{g})}{\partial
R}+\frac{\partial^{2}(\phi_{s}+\phi_{g})}{\partial z^{2}}=4\pi
G(\Sigma_{s}+\Sigma_{g})\delta(z)$ (5)

Here,$u_{i}$ and $v_{i}$ are the perturbed velocities in radial and tangential
direction, $\Sigma_{i}$ and $\phi_{i}$ are the perturbed surface densities and
the potentials of the $i^{th}$ components. The quantities $\Sigma_{0i}$ ,
$\Omega_{0}$ and $\phi_{0i}$ are the unperturbed surface densities, angular
velocity and potential respectively.

Considering axis-symmetric case $\frac{\partial}{\partial{\theta}}\rightarrow
0$, the perturbed equations read;

$\frac{\partial u_{i}}{\partial
t}-2\Omega_{0}v_{i}+\frac{c_{i}^{2}}{\Sigma_{0i}}\frac{\partial\Sigma_{i}}{\partial
R}+\frac{\partial(\phi_{s}+\phi_{g})}{\partial
R}+\frac{\Sigma_{i}}{\Sigma_{0i}}\frac{\partial\phi_{ext}}{\partial R}=0$ (6)

$\frac{\partial v_{i}}{\partial t}-2Bu_{i}=0$ (7)

$\frac{\partial\Sigma_{i}}{\partial t}+\Sigma_{0i}\frac{\partial
u_{i}}{\partial R}=0$ (8)

and the Poisson equation for the thin disc assumes the form Toomre (1964);

$\frac{\partial(\phi_{s}+\phi_{g})}{\partial R}=-2\pi
iG(\Sigma_{s}+\Sigma_{g})$ (9)

Equations 6,7,8,9 will be basis for deriving the dynamic equations governing
growth of perturbations and also the associated dispersion relation. Indexing
equation 6,7,8,9 for stars;

$\frac{\partial u_{s}}{\partial
t}-2\Omega_{0}v_{s}+\frac{c_{s}^{2}}{\Sigma_{0s}}\frac{\partial\Sigma_{s}}{\partial
R}+\frac{\partial(\phi_{s}+\phi_{g})}{\partial
R}+\frac{\Sigma_{s}}{\Sigma_{0s}}\frac{\partial\phi_{ext}}{\partial R}=0$ (10)

$\frac{\partial v_{s}}{\partial t}-2Bu_{s}=0$ (11)

$\frac{\partial\Sigma_{s}}{\partial t}+\Sigma_{0s}\frac{\partial
u_{s}}{\partial R}=0$ (12)

Combining the equations (9), (10), (11), and (12) we obtain the following
equation for stars;

$\frac{\partial^{2}\Sigma_{s}}{\partial
t^{2}}-c_{s}^{2}\frac{\partial^{2}\Sigma_{s}}{\partial
R^{2}}-4\Omega_{0}B\Sigma_{s}+2\pi iG\Sigma_{0s}\frac{\partial}{\partial
R}(\Sigma_{s}+\Sigma_{g})-\Sigma_{s}\frac{\partial^{2}\phi_{ext}}{\partial
R^{2}}=0$ (13)

similar equation for gas reads ;

$\frac{\partial^{2}\Sigma_{g}}{\partial
t^{2}}-c_{g}^{2}\frac{\partial^{2}\Sigma_{g}}{\partial
R^{2}}-4\Omega_{0}B\Sigma_{g}+2\pi iG\Sigma_{0g}\frac{\partial}{\partial
R}(\Sigma_{s}+\Sigma_{g})-\Sigma_{g}\frac{\partial^{2}\phi_{ext}}{\partial
R^{2}}=0$ (14)

In order to obtain a dispersion relation we substitute plane wave ansatz
$e^{i(kr-\omega t)}$ for the perturbed quantities in equations (13) and (14),
we obtain

$\Sigma_{s}=\frac{2\pi
Gk\Sigma_{0s}\Sigma_{g}}{(\omega^{2}-c_{s}^{2}k^{2}-\kappa^{2}+2\pi
G\Sigma_{0s}k-T_{0})}$ (15)

and similarly

$\Sigma_{g}=\frac{2\pi
Gk\Sigma_{0g}\Sigma_{s}}{(\omega^{2}-c_{g}^{2}k^{2}-\kappa^{2}+2\pi
G\Sigma_{0g}k-T_{0})}$ (16)

Where $T_{0}=-\frac{\partial^{2}\phi_{ext}}{\partial R^{2}}$. Combining
equation (15) and (16) the final dispersion relation reads;

$\displaystyle(\omega^{2}-c_{s}^{2}k^{2}-\kappa^{2}+2\pi
G\Sigma_{0s}k-T_{0})(\omega^{2}-c_{g}^{2}k^{2}-\kappa^{2}+2\pi
G\Sigma_{0g}k-T_{0})=$ (17) $\displaystyle(2\pi G\Sigma_{0s}k)(2\pi
G\Sigma_{0g}k)$

We now proceed to derive stability criterion a la Jog (1996). Defining the
following parameters;

$\displaystyle\alpha_{s}=\kappa^{2}+c_{s}^{2}k^{2}-2\pi G\Sigma_{0s}k+T_{0}$
(18) $\displaystyle\alpha_{g}=\kappa^{2}+c_{g}^{2}k^{2}-2\pi
G\Sigma_{0}k+T_{0}$ $\displaystyle\beta_{s}=2\pi G\Sigma_{0s}k$
$\displaystyle\beta_{g}=2\pi G\Sigma_{0g}k$

Substituting above the dispersion relation and the respective roots are given
by;

$\displaystyle\omega^{4}-\omega^{2}(\alpha_{s}+\alpha_{g})+(\alpha_{s}\alpha_{g}-\beta_{s}\beta_{g})=0$
(19)
$\displaystyle\omega^{2}_{\pm}=\frac{1}{2}(\alpha_{s}+\alpha_{g})\pm\frac{1}{2}((\alpha_{s}+\alpha_{g})^{2}-4(\alpha_{s}\alpha_{g}-\beta_{s}\beta_{g}))^{\frac{1}{2}}$

For a one component gaseous or stellar disc $\alpha_{g}\geq 0$ or
$\alpha_{s}\geq 0$ is sufficient condition for stability, whereas condition
for marginal stability of two-component disc reads $\omega^{2}=0$ or
$(\alpha_{s}\alpha_{g}-\beta_{s}\beta_{g})=0$. And for the disc to be unstable
the conditions is $\alpha_{s}\alpha_{g}-\beta_{s}\beta_{g}<0$. With simple
algebra the condition for neutral equilibrium can be written as;

$\frac{2\pi G\Sigma_{0s}k}{\kappa^{2}+k^{2}c_{s}^{2}+T_{0}}+\frac{2\pi
G\Sigma_{0g}k}{\kappa^{2}+k^{2}c_{g}^{2}+T_{0}}=F$ (20)

where F=1. For a marginally stable one fluid disc a function $\rm F$ can be
defined as $\rm F=\frac{2\pi G\Sigma_{0}k}{\kappa^{2}+k^{2}c^{2}+T_{0}}$, .
The value of $k_{min}$ for the 1 fluid disc is obtained by putting
$\frac{d\omega^{2}}{dk}=0$, where $\omega^{2}=\kappa^{2}-2\pi G\Sigma
k+c^{2}k^{2}+T_{0}$, which yields $k_{min}=\frac{\pi G\Sigma}{c^{2}}$.
Evaluating the polarisation function at $k_{min}$ yields
$F=\frac{2}{1+Q^{2}}$. In a manner similar to the one fluid case-fluid disc a
function $\rm F$ for the marginally stable two-fluid disc is evaluated at
$k_{min}$ which implies finding $\frac{d\omega^{2}_{-}}{dk}=0$,or
$\frac{d(\omega^{2}_{+}\omega^{2}_{-})}{dk}=0$, i.e finding
$\frac{d(\alpha_{s}\alpha_{g}-\beta_{s}\beta_{g})}{dk}$ which yields;

$\displaystyle k^{3}(4c_{s}^{2}c_{g}^{2})-3k^{2}(2\pi
G\Sigma_{0s}c_{s}^{2}+2\pi G\Sigma_{0g}c_{g}^{2})$ (21)
$\displaystyle+2k((\kappa^{2}+T_{0})(c_{g}^{2}+c_{s}^{2}))-(2\pi
G\Sigma_{0s}+2\pi G\Sigma_{0sg})(\kappa^{2}+T_{0})=0$

Now, defining $\epsilon=\frac{\Sigma_{0g}}{\Sigma_{0s}+\Sigma_{0sg}}$,and
parameter $X=\frac{\kappa^{2}}{2\pi G(\Sigma_{0s}+\Sigma_{0g})k_{min}}$, the
criterion for appraising the stability of two-fluid disc under influence of
external tidal field is given by;

$\frac{2}{1+Q_{T}^{2}}=\frac{(1-\epsilon)}{X(1+\frac{(1-\epsilon)^{2}Q_{s}^{2}}{4X^{2}}+\frac{T_{0}}{\kappa^{2}})}+\frac{\epsilon}{X(1+\frac{\epsilon^{2}Q_{g}^{2}}{4X^{2}}+\frac{T_{0}}{\kappa^{2}})}$
(22)

The above equation can also be written as;

$\frac{2}{1+Q_{T}^{2}}=\frac{(1-\epsilon)}{X^{\prime}(1+\frac{(1-\epsilon)^{2}Q^{\prime
2}_{s}}{4X^{\prime
2}})}+\frac{\epsilon}{X^{\prime}(1+\frac{\epsilon^{2}Q^{\prime
2}_{g}}{4X^{\prime 2}})}$ (23)

The primed quantities are defined as
$X^{\prime}=\frac{(\kappa^{2}+T_{0})}{2\pi G(\Sigma_{0s}+\Sigma_{0g})k_{min}}$
and $Q^{\prime}_{s}=\frac{\kappa c_{s}}{\pi
G\Sigma_{0s}}(1+\frac{T_{0}}{\kappa^{2}})^{\frac{1}{2}}$ and
$Q^{\prime}_{g}=\frac{\kappa c_{g}}{\pi
G\Sigma_{0g}}(1+\frac{T_{0}}{\kappa^{2}})^{\frac{1}{2}}$. The original
parameters describing the stability under influence of tidal field are
modified as $X^{\prime}=X(1+\frac{T_{0}}{\kappa^{2}})$,
$Q^{\prime}_{s}=Q_{s}(1+\frac{T_{0}}{\kappa^{2}})^{\frac{1}{2}}$ and
$Q^{\prime}_{g}=Q_{g}(1+\frac{T_{0}}{\kappa^{2}})^{\frac{1}{2}}$. Where the
stability parameter $Q_{T}$ is evaluated at $k_{min}$. $Q_{T}>1,=1,<1$
correspond to stable, marginally stable and unstable configurations of the
two-fluid disc.

##  3 Results/Discussion

To gain better insight into on impact of tidal forces on two fluid disc we
firstly investigate the impact of tidal force on a one-component disc. The
dispersion relation for a one-component disc in presence of tidal field is;

$\omega^{2}=(\kappa^{2}+T_{0})k^{(0)}+c^{2}k^{(2)}-2\pi G\Sigma k^{(1)}$ (24)

In the above equation at large value of $\rm k$, $k^{2}$ will dominate thus
presure stabilises the galactic disc at small scales, at small $\rm k$ i.e
$\rm k^{(0)}$ the differential rotation and the tidal field stabilise the disc
at large scales, and at intermdiate $\rm k$ self-gravity of the galactic disc
becomes important. Tidal field comes in two flavours ’compressive’ $\rm
T_{0}<0$ and ’disruptive’ $\rm T_{0}>0$. Classic example of disruptive tidal
field $\rm T_{0}>0$ is due to point mass. Renaud (2010), appendix B, has an
exhaustive compilation of tidal fields due to all the important potential-
density pairs. A disruptive tidal field adds up with the differential rotation
and presure to stabilise the disc, whereas a compressive tidal field adds up
with the self gravity of the galactic disc and aids in destabilising the
galactic disc. Next we proceed to inspect the marginal stability of the one-
fluid galactic disc. Putting $\omega^{2}=0$ in equation (25) can be recast to
obtain a quadratic equation in $\rm k$,

$1+\frac{Q^{\prime 2}}{4}\frac{k^{2}}{k^{\prime
2}_{T}}-\frac{k}{k^{\prime}_{T}}=0$ (25)

Where, $Q^{\prime}=Q(1+\frac{T_{0}}{\kappa^{2}})^{\frac{1}{2}}$,
$k^{\prime}_{T}=k_{T}(1+\frac{T_{0}}{\kappa^{2}})$,
$k_{T}=\frac{\kappa^{2}}{2\pi G\Sigma_{0}}$ and defining
$\zeta^{\prime}=\frac{k^{\prime}_{T}}{k}=\frac{k_{T}}{k}(1+\frac{T_{0}}{\kappa^{2}})$,
i.e $\zeta^{\prime}=\zeta(1+\frac{T_{0}}{\kappa^{2}})$. With above
substitutions equation (26) can be shown to assume;

$Q=2\left[\zeta\left(1-\zeta(1+\frac{T_{0}}{\kappa^{2}})\right)\right]^{\frac{1}{2}}$
(26)

Figure 1: The above figure depicts the marginal stability of the one fluid
disc under influence of tidal field. ’Solid’ line indicates marginal stabilty
in absence of tidal field, ’dashed’ line indicates marginal stabilty for
disruptive tidal field $T_{0}>0$ and ’dotted’ line indicates stabilty of disc
when acted upon by compressive tidal $T_{0}<0$.

The marginal stability curve (Fig 1) is plotted fixing
$\frac{T_{0}}{\kappa^{2}}$ at a fixed value of 0.5. It can be seen that for
for value of $Q<1$ in absence of tidal field the disc becomes unstable,
whereas disruptive tidal increase this value, so the disc is less succeptible
to external perturbations. And compressive tidal field lowers the value such
that the disc becomes more prone to axis-symmetric perturbations.

Now we will try to address the role of compressive and disruptive tidal fields
on a two fluid galaxy disc. We plot contours of $Q_{T}$, for the given values
of $Q_{s},Q_{g},\epsilon,$ and $\frac{T_{0}}{\kappa^{2}}$. For the same range
of values spanning $X_{s}=\frac{1}{2}(1-\epsilon)Q_{s}^{2}$ and
$X^{\prime}_{g}=\frac{1}{2}\epsilon Q_{g}^{2}$, and are computed over range of
$Q_{s}$ and $Q_{g}$ at a constant value of $\epsilon$ and
$\frac{T_{0}}{\kappa^{2}}$. $Q_{T}$ is evaluated by finding minimum $X_{min}$
and maximum $X_{max}$ in range between $X_{s}$ and $X_{g}$, and finally
$Q_{T}$ is evaluated over a range of $X_{max}$ and $X_{min}$, with the idea
that $X$ lies in between $X_{max}$ and $X_{min}$. Procedure and applications
of two-component stability formula without external tidal are discussed in
greater detail in Jog (1996). For studying impact of tidal force we plot
contour diagrams fixing gas-fraction $\epsilon$ and
$R=\frac{T_{0}}{\kappa^{2}}$ and for the same gas fraction we will vary $R$ to
understand the impact of the external force on two-fluid disc. In table (1)
and table (2) are summarised the values of maximum and minimum values of
$Q_{T}$ obtained for varying values of gas-fraction $\epsilon$ and strength of
external tidal field.

$R\rightarrow$ | 0.0 | +0.5 | +2.5 | -0.1 | -0.5
---|---|---|---|---|---
$\epsilon\downarrow$ |  |  |  |  |
0.05 | 1.794 | 2.098 | 2.693 | - | 1.44
0.1 | 1.675 | 1.88 | 2.515 | - | 1.280
0.3 | 1.306 | 1.599 | 2.55 | 1.88 | -
Table 1: Above table describes the values of maximum values of $Q_{T}$ obtained for two fluid galactic disc under influence of external tidal fields by varying $T_{0}$ and $\epsilon$ $R\rightarrow$ | 0.0 | +0.5 | +2.5 | -0.1 | -0.5
---|---|---|---|---|---
$\epsilon\downarrow$ |  |  |  |  |
0.05 | 0.994 | 1.198 | 1.693 | - | 0.647
0.1 | 0.875 | 0.81 | 1.515 | - | 0.480
0.3 | 0.406 | 0.699 | 1.134 | 0.388 | -
Table 2: Above table describes the values of mimimum values of $Q_{T}$
obtained for two fluid galactic disc under influence of external tidal fields
by varying $T_{0}$ and $\epsilon$
---
Figure 2: The above plots indicate contours of $Q_{T}$ plotted against $Q_{g}$
and $Q_{s}$, at constant value of $\epsilon=0.05$. Panel 1 indicates stability
when $R=0$, panel 2 for disruptive tidal field $\rm R=0.5$
---
Figure 3: The above plots indicate contours of $Q_{T}$ plotted against $Q_{g}$
and $Q_{s}$, at constant value of $\epsilon=0.05$. Panel 1 indicates stability
at $\rm R=+2.5$ and panel 2 for compressive tidal field $\rm(R=-0.5)$..
---
Figure 4: The above plots indicate contours of $Q_{T}$ plotted against $Q_{g}$
and $Q_{s}$, at constant value of $\epsilon=0.1$. Panel 1 indicates stability
when $R=0$, panel 2 for disruptive tidal field $\rm R=+0.5$
---
Figure 5: The above plots indicate contours of $Q_{T}$ plotted against $Q_{g}$
and $Q_{s}$, at constant value of $\epsilon=0.1$. Panel 1 inducates stability
at $\rm(R=+2.5)$ and panel 2 for compressive tidal field $\rm R=-0.5$
---
Figure 6: The above plots indicate contours of $Q_{T}$ plotted against $Q_{g}$
and $Q_{s}$, at constant value of $\epsilon=0.3$. Panel 1 indicates stability
when $\rm R=0$, panel 2 for disruptive tidal field $\rm R=+0.5)$.
---
Figure 7: The above plots indicate contours of $Q_{T}$ plotted against $Q_{g}$
and $Q_{s}$, at constant value of $\epsilon=0.3$. panel 1 indicates stability
for $\rm R=+2.5$ and panel 2 for compressive tidal field $\rm R=-0.1)$.

The main results of the work can be inferred directly from the table;
$\rm 1)$ Increasing gas fraction destabilises the disc,
$\rm 2)$ A disruptive tidal field tends to increase the stability of the disc
even at high gas-fraction, whereas compressive tidal field destabilises the
galaxy disc.

It can be see that by increasing the value of $\epsilon$ from 0.05 to 0.3
changes $Q_{Tmax}$ from 1.794 to 1.306. By applying a disruptive tidal field
of intermediate strength $R=+0.5$ the value of $Q_{Tmax}$ for $T_{0}=0$
increases from 1.79 to 2.098 and for an intense tidal field $R=2.5$ the
maximum value of stability $Q_{T}$ increases to 2.69, which is much higher
than either of just stellar or gas disc. Even at very high gas fraction when
the disc by itself is unstable the disruptive tidal forces tends to stabilise
the disc as an example (see table 2) at a gas fraction of 0.3 without tidal
forces the values of lowest stability contour is 0.406 and application of
$R=+2.5$ makes the disk just stable $Q_{Tmin}=1.012$. Compressive tidal force
has a destabilising effect on the galactic disc, it can be seen that even a
tidal field of intermediate strength $R=-0.1$ can destabilise the disk at high
gas fraction, see table 1, at gas fraction of about 0.3 when $R=0$,
$Q_{Tmax}=1.306$ and at $R=-0.1$, $Q_{Tmax}=1.188$. And similarly when $T=0$,
for gas fraction $\epsilon=0.3$, $Q_{Tmin}=0.406$ and under influence of an
external tidal field $R=-0.1$, $Q_{Tmin}=0.388$.

As a simple possible application we will study the impact of gas-fraction and
tidal forces on regulating the growth rate of the instabilities
($\omega_{-}^{2}$) in the two-fluid disc. The growth rate of instabilities in
two-fluid galactic disc is (see equation 19);

$\displaystyle\omega^{2}_{-}=\frac{1}{2}(\alpha_{s}+\alpha_{g})-\frac{1}{2}((\alpha_{s}+\alpha_{g})^{2}-4(\alpha_{s}\alpha_{g}-\beta_{s}\beta_{g}))^{\frac{1}{2}}$
(27)

where (see equation 18)

$\displaystyle\alpha_{s}=\kappa^{2}+c_{s}^{2}k^{2}-2\pi G\Sigma_{0s}k+T_{0}$
(28) $\displaystyle\alpha_{g}=\kappa^{2}+c_{g}^{2}k^{2}-2\pi
G\Sigma_{0}k+T_{0}$ $\displaystyle\beta_{s}=2\pi G\Sigma_{0s}k$
$\displaystyle\beta_{g}=2\pi G\Sigma_{0g}k$

The above quantities can be expressed as

$\displaystyle\frac{\alpha_{s}}{\kappa^{2}}=1+\frac{1}{4}Q_{s}^{2}(1-\epsilon)^{2}(\frac{k}{k_{T}})^{2}-(1-\epsilon)\frac{k}{k_{T}}+R$
(29)
$\displaystyle\frac{\alpha_{g}}{\kappa^{2}}=1+\frac{1}{4}Q_{g}^{2}\epsilon^{2}(\frac{k}{k_{T}})^{2}-\epsilon\frac{k}{k_{T}}+R$
$\displaystyle\frac{\beta_{s}}{\kappa^{2}}=(1-\epsilon)\frac{k}{k_{T}}$
$\displaystyle\frac{\beta_{g}}{\kappa^{2}}=\epsilon\frac{k}{k_{T}}$

Here, $k_{T}=\frac{\kappa^{2}}{2\pi G(\Sigma_{0s}+\Sigma_{0g})}$, for each
case we will fix the values of $Q_{s}$, $Q_{g}$ and the gas fraction
$\epsilon$ and vary the value of $R$ to see the effect of the tidal forces on
the two-fluid disc.

---
Figure 8: The above plots indicate the growth rate $\omega_{-}^{2}$ for
varying value of R= 0, +0.5, +2.5, -0.5, -2.5 at a constant value of
$\epsilon$=0.05 and $Q_{s}$=$Q_{g}$=1.
---
Figure 9: The above plots indicate the growth rate $\omega_{-}^{2}$ for
varying value of R= 0, +0.5, +2.5, -0.5, -2.5 at a constant value of
$\epsilon$=0.3 and $Q_{s}$=$Q_{g}$=1.
---
Figure 10: The above plots indicate the growth rate $\omega_{-}^{2}$ for
varying value of R= 0, +0.5, +2.5, -0.5, -2.5 at a constant value of
$\epsilon$=0.05 and $Q_{s}$=$Q_{g}$=2.
---
Figure 11: The above plots indicate the growth rate $\omega_{-}^{2}$ for varying value of R= 0, +0.5, +2.5, -0.5, -2.5 at a constant value of $\epsilon$=0.3 and $Q_{s}$=$Q_{g}$=2.  $R\rightarrow$ | 0.0 | +0.5 | 2.5 | -0.5 | -2.5
---|---|---|---|---|---
$\epsilon\downarrow$ |  |  |  |  |
0.05 | -0.232 | 0.267 | 2.26 | -0.73 | -2.73
0.3 | -2.35 | -1.86 | 0.140 | -2.85 | -4.85
Table 3: Above table depicts the values of $\omega_{-(min)}^{2}$ at constant value of $Q_{s}=Q_{g}=1$. $R\rightarrow$ | 0.0 | +0.5 | 2.5 | -0.5 | -2.5
---|---|---|---|---|---
$\epsilon\downarrow$ |  |  |  |  |
0.05 | 0.69 | 1.19 | 3.19 | 0.19 | -1.80
0.3 | 0.16 | 0.66 | 2.66 | -0.377 | -2.33
Table 4: Above table depicts the values of $\omega_{-(min)}^{2}$ at constant
value of $Q_{s}=Q_{g}=2$.
---
Figure 12: The above plots indicate the growth rate $\omega_{+}^{2}$ for
varying value of compressive tidal field R= -0.5, -2.5, -4.0 value of
$\epsilon$=0.3, 0.05 and $Q_{s}$=$Q_{g}$=1.

From Table 4 and Table 5 we can infer that a disruptive tidal fields tends to
makes $\omega_{-(min)}$ more positive,thus making disc less prone to growth of
instabilities, whereas a compressive tidal field tends to make
$\omega_{-(min)}$ more negative and thus making the disc more susceptible to
the growth of axis-symmetric perturbations. From table 4 we can see that
growth rate $\omega_{-(min)}$ is positive for $Q_{s}=Q_{g}=2$, but intense
compressive tidal of order R= -2.5 at even very small gas fraction can make
$\omega_{-(min)}$ negative. The growth rate of instabilities is enhanced
immensely in case of galactic disc with high gas fraction aided by external
compressive tidal field. Finally from figure 12 it can be seen that under
influence of intense compressive tidal fields even the $\omega_{+(min)}$ can
become negative thus further enhancing the growth of instabilities in galactic
disc.

##  4 Conclusion

In this work differential equations governing the growth of instabilities in
two-fluid galactic disc under external tidal field have been derived and a
criterion for appraising the stability of the two-fluid disc is presented. The
modified stability criterion is applied to the understand the influence of the
compressive and disruptive tidal fields, along with role of gas fraction on
the stability of the galactic disc. We have also derived modified dispersion
relation and studied the effect of the tidal fields on growth rate of
instabilites in galactic disc, and interestingly found out that the
$\omega_{+}^{2}$ can also become negative and can further destabilise the
disc. Some possible applications of the results presented here are:
understanding the role of tidal fields in enhancing or quenching of star
formation rates in interacting galaxies, understanding the role of dark matter
haloes in stabilising gas rich low surface brightness galaxies,which without
the tidal effects of the dark matter haloes will be unstable owing to large
gas-fraction.

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