Open Access

Jackson’s \((-1)\)-Bessel functions with the Askey-Wilson algebra setting

Advances in Difference Equations20152015:268

https://doi.org/10.1186/s13662-015-0612-6

Received: 18 January 2015

Accepted: 19 August 2015

Published: 29 August 2015

Abstract

This work is devoted to the study of some functions arising from a limit transition of the Jackson q-Bessel functions when q tends to −1. These functions coincide with the so-called cas function for particular values of parameters. We prove that there are eigenfunctions of differential-difference operators of Dunkl-type. Also we consider special cases of the Askey-Wilson algebra \(AW(3)\), which have these operators (up to constants) as one of their three generators and whose defining relations are given in terms of anti-commutators.

Keywords

q-special functionsdifference-differential equationsAskey-Wilson algebra

MSC

33D4533D80

1 Introduction

In [1], Vinet and Zhedanov introduced new families of orthogonal polynomials by considering appropriate limits when q tends to −1 of the little and big q-Jacobi polynomials. In this work we will study the limit when q tends to −1 of the three q-analogs of the Bessel function, which are introduced by Jackson [24]. The first and the second q-Bessel functions are reconsidered and rewritten in modern notations by Ismail [5]. The third q-Bessel function is rediscovered later by Hahn [6] and Exton [7]. This function has an interpretation as matrix elements of irreducible representations of the quantum group of plane motions \(E_{q}(2)\) and satisfies an orthogonality relation that makes it more suitable for harmonic analysis [810]. Of course, when q tends to 1, the Jackson q-Bessel functions tend to the standard Bessel function [11].

In this paper we will show that the limit when q tends to −1 of the third q-Bessel functions leads to a new type of nonsymmetric Bessel functions satisfying first order differential-difference equation. Also these functions coincide for a particular value of its parameters with the cas function [12]. Furthermore, by using the limit transition from little q-Jacobi polynomials to the third q-Bessel function and from q-Laguerre to the second q-Bessel function we construct a q-Bessel version of the Askey-Wilson \(AW(3)\) algebra.

Notations

Throughout we assume \(-1< q<1\). For q-Pochhammer symbols and q-hypergeometric series we use the notation of [13],
$$ (a ; q)_{0}: =1 ,\qquad (a ; q)_{n} := \prod _{k=0}^{n-1} \bigl(1 - aq^{k} \bigr),\quad n=1, \ldots, \infty . $$
(1.1)
The basic hypergeometric series are defined by
$$ {}_{r}\Phi_{s} \biggl( \begin{matrix} a_{2}, \ldots, a_{r}\\ b_{1},\ldots,b_{s} \end{matrix} \Big| q; z \biggr):= \sum_{k=0}^{+\infty} \frac{ (a_{1},\ldots, a_{r} ; q )_{k}}{ (q,b_{1},\ldots, b_{s} ; q )_{k}} \bigl((-1)^{k}q^{ (^{k}_{2} )} \bigr)^{1+s-r}z^{k}, $$
(1.2)
where
$$ (a_{1}, \ldots, a_{r} ; q )_{n}:= (a_{1}; q )_{n}\cdots (a_{r}; q )_{n}. $$

2 Askey-Wilson relations for the little q-Jacobi polynomials

2.1 Little q-Jacobi polynomials

The little q-Jacobi polynomial is defined by [14], (18.27(iv))
$$ p_{n}(x) =p_{n}(x;a,b | q):={}_{2}\Phi_{1} \biggl( \begin{matrix} q^{-n}, abq^{n+1}\\ aq \end{matrix} \Big| q; qx \biggr). $$
(2.1)
For \(0< a< q^{-1}\) and \(b< q^{-1}\), the polynomials \(\{p_{n}(x)\}_{n}\) satisfy the orthogonality relations
$$\frac{(aq;q)_{\infty}}{(abq^{2};q)_{\infty}}\sum_{k=0}^{\infty} p_{m}\bigl(q^{k}\bigr)p_{n}\bigl(q^{k} \bigr) (aq)^{k}\frac{(bq;q)_{k}}{(q;q)_{k}} = \frac{(1-abq)(aq)^{n}}{1-abq^{2n+1}}\frac {(q,bq;q)_{n}}{(aq,abq;q)_{n}} \delta_{m,n}. $$
There is a q-difference equation for this polynomial of the form
$$ Y_{a,b,q}f(x)=\lambda_{n} f(x), $$
(2.2)
where
$$ (Y_{a,b,q}f):=a\bigl(bq-x^{-1}\bigr) \bigl(f(qx)-f(x)\bigr)+ \bigl(1-x^{-1}\bigr) \bigl(f\bigl(q^{-1}x\bigr)-f(x)\bigr) $$
(2.3)
and
$$ \lambda_{n}=q^{-n}\bigl(1-q^{n}\bigr) \bigl(1-abq^{n+1}\bigr). $$
(2.4)
The Askey-Wilson algebra \(AW(3)\) involves a nonzero scalar q and three parameters \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\), it was introduced by Zhedanov [15] as an associative algebra generated by X, Y, and Z subject to the following commutation relations:
$$ \begin{aligned} &YX -q XY = \mu_{3}Z + \omega_{3},\qquad ZY -qYZ = \mu_{2}X + \omega_{2},\\ & XZ -qZXY = \mu_{1}Y + \omega_{1}. \end{aligned} $$
(2.5)
There is a central element Q, which is explicitly given as a polynomial of degree 3 in terms of X, Y, and Z [15]
$$ Q = \bigl(q^{2} - 1\bigr)YXZ + \mu_{1}Y^{2} + \mu_{2}qX^{2} + \mu_{3}Z^{2} + (q + 1) ( \omega_{1}Y + \omega_{2}qX + \omega_{3}Z). $$
(2.6)
The limit of orthogonal polynomials in the Askey scheme as \(q\rightarrow1\) corresponds to the limit \(q \rightarrow1\) of \(AW(3)\) to some classical algebras. In particular, the related Askey-Wilson algebra for the little q-Jacobi polynomial is generated by X, Y, Z with relations [1]
$$\begin{aligned} YX -qXY = Z + \omega_{3},\qquad ZY -qYZ = X + \omega_{2},\qquad XZ -qZX =0, \end{aligned}$$
(2.7)
where
$$ \omega_{2}=-\frac{1+b}{(1+q)b},\qquad \omega_{3}=- \frac{1+a}{(1+q)\sqrt {ab}}. $$
(2.8)
There is a representation on the space of polynomials of the little q-Jacobi \(AW(3)\) defined by relations (2.7) with structure constants given in (2.8) as follows:
$$ \begin{aligned} &(Yf) (x):=\mu(Y_{a,b,q}+1+qab)f(x),\qquad (Xf) (x):=xf(x),\\ & (Zf) (x):= \frac{1-x}{q\sqrt{ab}}f\bigl(q^{-1}x\bigr), \end{aligned} $$
(2.9)
where
$$\mu=-\frac{1}{(1-q^{2})\sqrt{ab}}. $$
The Casimir operator
$$ Q = \bigl(q^{2} - 1\bigr)YX Z + q^{2}X^{2} + Z^{2} + (q + 1) (\omega_{2}qX + \omega_{3}Z) $$
(2.10)
takes the value \(Q=-b^{-1}\).

2.2 Little \((-1)\)-Jacobi polynomials

The little \((-1)\)-Jacobi polynomials \(P_{n}^{(\alpha, \beta,-1)}(x)\) have been introduced and investigated in [1] as limits of the little q-Jacobi polynomials (2.1)
$$ \lim_{\varepsilon\rightarrow0}p_{n}\bigl(x;-e^{\varepsilon \alpha},-e^{\varepsilon\beta} | -e^{\varepsilon}\bigr)=P_{n}^{(\alpha, \beta,-1)}(x). $$
(2.11)
We recall here their basic properties [1]. The polynomial \(P_{n}^{(\alpha, \beta,-1)}(x)\) satisfies the following difference-differential equation
$$ (Y_{\alpha ,\beta ,-1}f) (x)=\lambda_{n}f(x), $$
(2.12)
where
$$ (Y_{\alpha ,\beta ,-1}f) (x)=(x-1)f^{\prime}(-x)+\bigl(\alpha+\beta-\alpha x^{-1}\bigr)\frac{f(x)-f(-x)}{2} $$
and
$$ \lambda_{n}=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} - n, &\mbox{if } n \mbox{ is even}, \\ \alpha+\beta+n+1 , &\mbox{if } n \mbox{ is odd}. \end{array}\displaystyle \right . $$
These polynomials have the following expressions in terms of the hypergeometric series:
For n, even
$$\begin{aligned} &P_{n}^{(\alpha, \beta,-1)}(x)= {}_{2}F_{1} \left ( \begin{matrix} -\frac{n}{2}, \frac{\alpha+\beta +n+2}{2} \\ \frac{\alpha+1}{2} \end{matrix} ;x^{2} \right )+\frac{nx}{\alpha+1} {}_{2}F_{1}\left ( \begin{matrix} 1-\frac{n}{2}, \frac{\alpha+\beta +n+2}{2} \\ \frac{\alpha+3}{2} \end{matrix} ;x^{2} \right ), \end{aligned}$$
and for n odd
$$\begin{aligned} P_{n}^{(\alpha, \beta,-1)}(x)= {}_{2}F_{1}\left ( \begin{matrix} -\frac{n-1}{2}, \frac{\alpha+\beta+ n+1}{2} \\ \frac{\alpha+1}{2} \end{matrix} ;x^{2} \right )-\frac{(\alpha+\beta+n+1)x}{\alpha+1}\ {}_{2}F_{1}\left ( \begin{matrix} -\frac{n-1}{2}, \frac{\alpha+\beta +n+3}{2} \\ \frac{\alpha+3}{2} \end{matrix} ;x^{2} \right ). \end{aligned}$$
We introduce the operators
$$\begin{aligned} \begin{aligned} &(Y f) (x):=(\mathrm{Y}_{\alpha ,\beta ,-1}f) (x)-\frac{1}{2}(\alpha+\beta+1)f(x),\qquad (Xf) (x):=xf(x),\\ & (Zf) (x):=(x-1)f(-x). \end{aligned} \end{aligned}$$
In [1], it was shown that these operators are closed in the framework of the Askey-Wilson algebra and they satisfy the commutation relations
$$\begin{aligned} \{X,Y\}=Z + \alpha, \qquad\{X,Z\}=0, \qquad\{Y,Z\}=Y+\beta, \end{aligned}$$
(2.13)
with \(\{A,B\} = AB +BA\) denoting as usual the anti-commutator of A and B.
The Casimir operator is
$$ Q =Y^{2}+ Z^{2} $$
(2.14)
and takes the value
$$Q=1. $$

3 The nonsymmetric Hankel transform

For Bessel functions \(J_{\alpha }(x)\) see [14], Chapter 10, and the references given therein. Let us consider the normalized Bessel function \(\mathcal{J}_{\alpha }(x)\), which is given by
$$ \mathcal{J}_{\alpha }(x):=\Gamma (\alpha +1) (2/x)^{\alpha }J_{\alpha }(x). $$
(3.1)
Then
$$\mathcal{J}_{\alpha }(x)= \sum_{k=0}^{\infty }\frac{(-\frac{1}{4} x^{2})^{k}}{(\alpha +1)_{k} k!} ={}_{0}F_{1} \biggl( \begin{matrix} {-}\\ {\alpha +1} \end{matrix}; -\frac{1}{4}x^{2} \biggr) \quad(\alpha >-1). $$
\(\mathcal{J}_{\alpha }(x)\) is an entire function and has the simple properties and special cases [14], Section 10.16.9
$$\mathcal{J}_{\alpha }(x)=\mathcal{J}_{\alpha }(-x),\qquad \mathcal{J}_{\alpha }(0)=1, \qquad\mathcal{J}_{-1/2}(x)=\cos x,\qquad \mathcal{J}_{1/2}(x)=\frac{\sin x}{x} . $$
The function \(x\mapsto\mathcal{J}_{\alpha }(\lambda x)\) satisfies also the eigenvalue equation [14], Section 10.13.5:
$$\biggl(\frac{d^{2}}{dx^{2}}+\frac{2\alpha +1}{x} \frac{d}{dx} \biggr) \mathcal{J}_{\alpha }(\lambda x)=-\lambda ^{2} \mathcal{J}_{\alpha }( \lambda x). $$
The Hankel transform pair [14], Section 10.22(v), for f in a suitable function class, is given by
$$ \left \{ \textstyle\begin{array}{@{}l} \widehat{f}(\lambda)=\int_{0}^{\infty} f(x)\mathcal{J}_{\alpha }(\lambda x)\frac{x^{2\alpha+1}\,dx}{2^{\alpha+1/2}\Gamma(\alpha+1)} ,\\ f(x)=\int_{0}^{\infty} \widehat{f}(\lambda) \mathcal{J}_{\alpha }(\lambda x)\frac{\lambda^{2\alpha+1}\,d\lambda }{2^{\alpha +1/2}\Gamma(\alpha+1)} . \end{array}\displaystyle \right . $$
(3.2)
Now consider the so-called nonsymmetric Bessel function, also called Dunkl-type Bessel function, in the rank one case (see [16], Section 4):
$$ \mathcal{E}_{\alpha }(x):=\mathcal{J}_{\alpha }(x)+\frac{i x}{2(\alpha +1)} \mathcal {J}_{\alpha +1}(x). $$
(3.3)
In particular,
$$\mathcal{E}_{-1/2}(x)=e^{ix}. $$
The nonsymmetric Hankel transform pair takes the form
$$ \left \{ \textstyle\begin{array}{@{}l} \widehat{f}(\lambda)=\int_{\mathbb {R}} f(x)\mathcal{E}_{\alpha }(\lambda x)\frac {|x|^{2\alpha+1}\,dx}{2^{\alpha+1}\Gamma(\alpha+1)} ,\\ f(x)=\int_{\mathbb {R}} \widehat{f}(\lambda) \mathcal{E}_{\alpha }(\lambda x)\frac{|\lambda|^{2\alpha+1}\,d\lambda }{2^{\alpha+1}\Gamma(\alpha+1)} . \end{array}\displaystyle \right . $$
(3.4)
The transform pair (3.4) follows immediately from (3.2). For some given α let us define the differential-reflection operator
$$ (T_{\alpha }f) (x):=f'(x)+\biggl(\alpha +\frac {1}{2}\biggr) \frac{f(x)-f(-x)}{x}, $$
(3.5)
called the Dunkl operator for root system \(A_{1}\) (see [17], Definition 4.4.2). We have the eigenvalue equation
$$ T_{\alpha }\bigl(\mathcal{E}_{\alpha }(\lambda x)\bigr)=i \lambda \mathcal{E}_{\alpha }(\lambda x ) . $$
(3.6)
If in (3.6) we substitute (3.3), compare even and odd parts, and then substitute (3.1), then we see that (3.6) is equivalent to a pair of lowering and raising differentiation formulas for Bessel functions (see [14], (10.6.2)):
$$J_{\alpha }'(x)- \frac{\alpha }{x} J_{\alpha }(x)=-J_{\alpha +1}(x),\qquad J_{\alpha +1}'(x)+\frac{\alpha +1}{x} J_{\alpha +1}(x)=J_{\alpha }(x). $$
The double degeneration of the double affine Hecke algebra \(\mathcal{H}"\) is generated by D, Z, and s with relations [18]
$$ sZs^{-1}=-Z,\qquad sDs^{-1}=-D,\qquad [D,Z]=1+2ks. $$
(3.7)
The operators D, s, and Z act on the polynomial \(f(x)\) as
$$ (D f) (x)=(T_{\alpha}f) (x),\qquad (sf) (x)=f(-x),\qquad (Zf) (x)=xf(x). $$
(3.8)
The operators defined in (3.8) are also known as para-Bose operators and the algebra (3.7) is equivalent to the para-Bose algebra [19, 20].
Another important generalization of the exponential function is the so-called cas function, which is given by [12, 21]
$$ \operatorname{cas}(x)=\cos(x)+\sin(x). $$
(3.9)
It is evident that the function \(y(x)=\operatorname{cas}(\lambda x)\) satisfies
$$ (\Lambda y) (x)=\lambda y(x),\qquad y(0)=1, $$
(3.10)
where \(\Lambda=s\partial\) and is the derivative operator.

4 Limit \(q\rightarrow-1\) of the third q-Bessel function

The third q-Bessel function was introduced in [6, 7], see also [7, 9, 10], and is defined by
$$\begin{aligned} J^{(3)}_{\nu}(x;q):=\frac{(q^{\nu+1};q)_{\infty }}{(q;q)_{\infty }} \biggl( \frac {1}{2}x\biggr)^{\nu}{}_{1}\Phi_{1} \biggl( \begin{matrix} 0\\ q^{\nu+1} \end{matrix} \Big| q; \frac{1}{4} qx^{2} \biggr) \quad(x>0). \end{aligned}$$
We will consider a slightly different function \(\mathcal{J}_{3}(x,a;q)\), called a normalized third q-Bessel function, which is defined by
$$\begin{aligned} \mathcal{J}_{3}(x,a;q)= {}_{1}\Phi_{1} \biggl( \begin{matrix} {0}\\ {aq} \end{matrix} \Big| q; qx \biggr). \end{aligned}$$
(4.1)
It is easy to see that
$$\lim_{q\rightarrow1}\mathcal{J}_{3} \bigl((1-q)^{2}x,q^{\alpha };q \bigr)=\mathcal{J}_{\alpha }(2\sqrt{x}). $$
The function \(\mathcal{J}_{3}(\lambda x,a;q)\) is a solution of the q-difference
$$\begin{aligned} (Y_{a,q}f) (x)=-\lambda f(x), \end{aligned}$$
(4.2)
where
$$ (Y_{a,q}f) (x):=\frac{a}{x} \bigl(f(qx)-f(x) \bigr)+ \frac{1}{x} \bigl(f\bigl(q^{-1}x\bigr)-f(x) \bigr). $$
(4.3)
Next, we describe the construction of new function by the limiting process from normalized third q-Bessel as \(q\rightarrow-1\) and from the \((-1)\)-Jacobi polynomials as \(n\rightarrow\infty\). Moreover, we have the following diagram for limit relations between these special functions and orthogonal polynomials.

The upper limit in the previous diagram is studied by [1]. Next we are concerned only by the three other limits.

There is a well-known limit from Jacobi polynomials to Bessel functions, see [14], (18.11.5),
$$ \lim_{n\to \infty }\frac{\Gamma (\alpha +1)}{n^{\alpha }} P^{(\alpha ,\beta )}_{n} \biggl(1-\frac{x^{2}}{2n^{2}} \biggr)=\mathcal{J}_{\alpha }( x). $$
(4.4)
The q-analog of this limit transition starts with the little q-Jacobi polynomials (2.1). From Proposition A.1 in [9] we have
$$ \lim_{n \rightarrow \infty}p_{n}\bigl(q^{n}x;a,b | q\bigr)= \mathcal{J}_{3}(x,a;q). $$
(4.5)
The operators X, Y, and Z defined in (2.9) have also a limit for \(n \rightarrow \infty \) after the rescaling \(x \rightarrow q^{n}x\). More precisely, let us denote
$$\begin{aligned} &(Xf) (x):=\lim_{n\rightarrow\infty}q^{-n}(Xf) \bigl(q^{n}x\bigr), \\ & (Yf) (x):=\lim_{n\rightarrow\infty}\sqrt{ab}q^{n}(Yf) \bigl(q^{n}x\bigr), \\ & (Zf) (x):=\lim_{n\rightarrow\infty}\sqrt{ab}(Zf) \bigl(q^{n}x \bigr). \end{aligned}$$
Then in the limit the operators X, Y, and Z are given by
$$\begin{aligned} (Xf) (x)=xf(x),\qquad (Yf) (x)=\frac{1}{q^{2}-1}(Y_{a,q}f) (x),\qquad (Zf) (x)=q^{-1}f\bigl(q^{-1}x\bigr), \end{aligned}$$
(4.6)
and (2.13) become
$$\begin{aligned} YX -qXY = Z + \omega_{3},\qquad ZY -qYZ= 0,\qquad XZ -qZX=0, \end{aligned}$$
(4.7)
where
$$ \omega_{3}=-\frac{1+a}{1+q}. $$
(4.8)
The resulting algebra generated by X, Y, and Z with (4.7) and (4.8) is called the q-Bessel \(AW(3)\) algebra. In this case the Casimir operator becomes
$$ Q = \bigl(q^{2} - 1\bigr)YXZ+ Z^{2} - (1+a)Z $$
(4.9)
and takes the value \(Q=-a\).
Equations (4.7) hold in the limit \(q\rightarrow-1\). Indeed, let us take the parametrization
$$a=-e^{\varepsilon(2 \alpha+1)} \quad \mbox{and}\quad q=-e^{\varepsilon}. $$
Since
$$\begin{aligned} &f\bigl(-e^{\pm\varepsilon}z\bigr)=f(-z)\mp zf(-z)\varepsilon+ \mathrm{o} (\varepsilon),\\ & q=-1-\varepsilon+\mathrm{o}(\varepsilon),\\ & a=-1-(2 \alpha+1)\varepsilon+\mathrm{o}(\varepsilon), \end{aligned}$$
the q-difference equation (4.2) tends formally as \(\varepsilon\rightarrow0\) to the differential-difference equation
$$ (Y_{\alpha }f) (x)=\lambda f(x), $$
(4.10)
where
$$ (Y_{\alpha }f) (x)=f^{\prime}(-x)+\biggl(\alpha+\frac{1}{2} \biggr)\frac {f(x)-f(-x)}{x} . $$
(4.11)
The operator \(Y_{\alpha }\) is a difference-differential operator of the first order containing reflection terms. Notice the important property of the operator \(Y_{\alpha }\): it sends the linear space of polynomials of dimension \(n+1\) to the space of dimension n. In particular, this means that there are no polynomial eigenfunction of the operator \(Y_{\alpha }\).
Now, let us introduce the operators
$$ (Xf) (x):=xf(x),\qquad (Y f) (x):=(Y_{\alpha}f) (x),\qquad (Zf) (x):=f(-x). $$
(4.12)
Then it is elementary to verify that the operators X, Y, Z satisfy the relations
$$\begin{aligned} \{X,Y\}=-Z - 2\alpha-1,\qquad \{X,Z\}=0,\qquad \{Y,Z\}=0, \end{aligned}$$
(4.13)
which corresponds to the \(AW(3)\) algebra with parameters
$$q = -1,\qquad \omega_{3} =-2\alpha -1,\qquad \mu_{3}=-1,\qquad \omega_{2} =\omega_{3} =\mu _{1}= \mu_{2}=0. $$
It is easily verified that the Casimir operator is
$$ Q = Z^{2}. $$
(4.14)
In the case of the realization (4.12) of the operators X, Y Z, the Casimir operator becomes the identity operator.

Theorem 4.1

For each \(\lambda\in\mathbb{C}\). The differential-difference equation (4.10) under the initial condition \(y(0)=1\), admits unique \(C^{\infty}\)-solution denoted \(\mathcal{J}_{\alpha,-1}(\lambda x)\), which is expressed in terms of the normalized q-Bessel function (3.1) by
$$ \mathcal{J}_{\alpha,-1}(\lambda x)=\mathcal{J}_{\alpha}(\lambda x)+ \frac{\lambda x}{2(\alpha+1)} \mathcal{J}_{\alpha+1}(\lambda x). $$
(4.15)

Proof

From the decomposition in the form \(f = f_{1} + f_{2}\) where \(f_{1}\) is even and \(f_{2}\) is odd. Equation (3.1) is equivalent to the following system:
$$ \left \{ \textstyle\begin{array}{@{}l} -f^{\prime}_{1}(x)=\lambda f_{2}(x), \\ f^{\prime}_{2}(x)+\frac{2\alpha+1}{x}f_{2}(x)=\lambda f_{1}(x),\\ f_{1}(0)=1,\qquad f^{\prime}_{1}(0)=0. \end{array}\displaystyle \right . $$
Thus
$$ \left \{ \textstyle\begin{array}{@{}l} -f^{\prime}_{1}(x)=\lambda f_{2}(x), \\ f^{\prime\prime}_{1}(x)+\frac{2\alpha+1}{x}f^{\prime}_{1}(x)=-\lambda ^{2} f_{1}(x),\\ f_{1}(0)=1,\qquad f^{\prime}_{1}(0)=0. \end{array}\displaystyle \right . $$
Hence,
$$ f_{1}(x)=\mathcal{J}_{\alpha}(\lambda x) \quad\mbox{and}\quad f_{2}(x)=-\frac{1}{\lambda}\frac{d}{dx}\mathcal{J}_{\alpha}( \lambda x). $$
 □
In particular, for \(\alpha=-1/2\), the \((-1)\)-Bessel function \(\mathcal{J}_{\alpha,-1}(\lambda x)\) coincides with the cas function (3.9)
$$\mathcal{J}_{-1/2,-1}(\lambda x)=\operatorname{cas}(\lambda x). $$
The function \(\mathcal{J}_{\alpha,-1}(\lambda x)\) is related to the Dunkl function (3.3) by
$$ \mathcal{J}_{\alpha,-1}(\lambda x)=\frac{1}{2}\bigl((1+i)\mathcal {E}_{\alpha }(\lambda x)+(1-i)\mathcal{E}_{\alpha}(-\lambda x)\bigr). $$
(4.16)
The \((-1)\)-Bessel transform pair takes the form
$$ \left \{ \textstyle\begin{array}{@{}l} \widehat{f}(\lambda)=\frac{1}{2^{\alpha+1}\Gamma(\alpha+1)} \int_{-\infty}^{\infty} f(x)\mathcal{J}_{\alpha,-1}(-\lambda x) |x|^{2\alpha+1}\,dx ,\\ f(x)=\frac{1}{2^{\alpha+1}\Gamma(\alpha+1)}\int_{-\infty}^{\infty} \widehat{f}(\lambda) \mathcal{J}_{\alpha,-1}(\lambda x) |\lambda|^{2\alpha+1}\,d\lambda. \end{array}\displaystyle \right . $$
(4.17)
The transform pair (4.17) follows immediately from (3.2) by putting \(f(x)=f_{1}(x)+x f_{2}(x)\) in (4.17) with \(f_{1}\) and \(f_{2}\) even.
The \((-1)\)-Bessel function \(\mathcal{J}_{\alpha,-1}(\lambda x)\) can be obtained also as limit case of the normalized third q-Bessel function (4.1)
$$ \lim_{\varepsilon \rightarrow 0}\mathcal{J}_{3}\bigl(x,-e^{\varepsilon(2\alpha+1)};-e^{\varepsilon} \bigr)= \mathcal{J}_{\alpha,-1}(x). $$
(4.18)
Indeed, from (4.1) we can expand \(\mathcal {J}(x,-e^{\varepsilon(2\alpha+1)};-e^{\varepsilon})\) in a power series of x as follows:
$$\mathcal{J}\bigl(x,-e^{\varepsilon(2\alpha+1)};-e^{\varepsilon}\bigr)=\sum _{n=0}^{\infty}c_{n,\alpha}(\varepsilon)x^{n}, $$
where
$$c_{n ,\alpha}(\varepsilon)=(-1)^{n}\frac{(-1)^{\frac {n(n-1)}{2}}(1-e^{2\varepsilon})^{n}}{4^{n}(e^{\varepsilon(2\alpha +2)},-e^{\varepsilon};-e^{\varepsilon})_{n}}. $$
Then the result follows from the following elementary limits involving q-shifted factorials:
$$\begin{aligned} &\lim_{\varepsilon\rightarrow0}\varepsilon^{-[n/2]}\bigl(-e^{\varepsilon \alpha}; -e^{\varepsilon}\bigr)_{n}=(-1)^{[n/2]}2^{n}\bigl(( \alpha+1)/2\bigr)_{[n/2]},\\ & \lim_{\varepsilon\rightarrow0}\varepsilon ^{-[(n+1)/2]}\bigl(e^{\varepsilon \alpha} ;-e^{\varepsilon}\bigr)_{n}=(-1)^{([n+1)/2]}2^{n}( \alpha/2)_{[(n+1)/2]}. \end{aligned}$$

5 The second q-Bessel function case

5.1 q-Laguerre polynomials

The q-Laguerre polynomials \(\{L_{n}(x,a;q)\}_{n}\) are defined by
$$L_{n}(x,a;q):=\frac{(aq;q)_{n}}{(q;q)_{n}} _{1}\phi_{1} \left ( \begin{matrix} q^{-n}\\ aq \end{matrix} \Big| q, -aq^{n+1} x \right ), $$
(5.1)
we have used slightly different notations (see [14], (18.27.15)). They satisfy the recurrence relations
$$\begin{aligned} -aq^{2n+1}xL_{n}(x,a;q)={}&\bigl(1-q^{n+1} \bigr)L_{n+1}(x,a;q)-\bigl[\bigl(1-q^{n+1}\bigr) \\ &{}+ q\bigl(1-aq^{n}\bigr)\bigr]L_{n}(x,a;q)+q \bigl(1-aq^{n}\bigr)L_{n-1}(x,a;q). \end{aligned}$$
(5.2)
There is a q-difference equation of the form
$$ (L_{a,q}y) (x)=-a\bigl(1-q^{n}\bigr)y(x), $$
(5.3)
where
$$ (L_{a,q}y) (x):=a\bigl(1+x^{-1}\bigr)y(qx)- \bigl[x^{-1}+a\bigl(1+x^{-1}\bigr)\bigr]y(x)+x^{-1}y \bigl(q^{-1}x\bigr). $$
(5.4)
When \(q\rightarrow1\) (\(a=q^{\alpha}\)) the q-Laguerre polynomial \(L_{n}(x,a;q)\) becomes the ordinary Laguerre polynomial
$$ L^{\alpha }_{n}(x):=\frac{(\alpha +1)_{n}}{n!} {}_{1}F_{1} \left (\begin{matrix} -n\\ \alpha +1 \end{matrix} \Big| x \right ). $$
(5.5)
There is a limit transition from little q-Jacobi to q-Laguerre (see [22], Section 4.12.2),
$$ L_{n}(x,a;q)=\lim_{b\rightarrow \infty }\frac{(qa;q)_{\infty }}{(q;q)_{\infty }} p_{n}\biggl(-\frac{x}{qb},a,b \Big| q\biggr). $$
(5.6)
Starting with the operators X, Y, and Z given by (2.9) we can also obtain the following operators:
$$\begin{aligned} &(Xf) (x):=-q\lim_{b \rightarrow \infty }b(Xf) (-x/qb),\\ & (Yf) (x):=\sqrt{a}\lim_{b \rightarrow \infty }\frac{1}{\sqrt {b}}(Yf) (-x/qb),\\ & (Zf) (x):=-q\lim_{b \rightarrow \infty }\sqrt{ab}(Zf) (-x/qb). \end{aligned}$$
Then
$$ \begin{aligned} &(Xf) (x):=xf(x),\\ &(Yf) (x):=\frac{q}{q^{2}-1}(L_{a,q}+a)f(x),\\ &(Zf) (x):=-f\bigl(q^{-1}x\bigr), \end{aligned} $$
(5.7)
where the operator \(L_{a,q}\) is defined in (5.4). A simple computation shows that the operators X, Y, Z satisfy the relations
$$\begin{aligned} &YX -qXY = Z + \omega_{3},\qquad ZY -qYZ = \omega_{2},\qquad XZ -qZX =0, \end{aligned}$$
(5.8)
where
$$\omega_{2}=\frac{aq}{1+q},\qquad \omega_{3}= \frac{q(1+a)}{1+q}. $$
(5.9)
The Casimir operator
$$ Q = \bigl(q^{2} - 1\bigr)YXZ + Z^{2} + (q + 1) ( \omega_{2}qX + \omega_{3}Z) $$
(5.10)
takes the value \(Q=-aq^{2}\).

5.2 Second Jackson’s q-Bessel function

The second Jackson’s q-Bessel function is defined as follows:
$$\begin{aligned} J^{(2)}_{\nu}(x;q):=\frac{(q^{\nu+1};q)_{\infty }}{(q;q)_{\infty }} \biggl( \frac {1}{2}x\biggr)^{\nu}{}_{0}\Phi_{1} \biggl( \begin{matrix} 0\\ q^{\nu+1} \end{matrix} \Big| q; -\frac{1}{4} q^{\nu+1}x^{2} \biggr)\quad (x>0). \end{aligned}$$
This notation is from [5] and is deferent from Jackson’s notation [24]. The classical Bessel function \(J_{\nu}\) is recovered by letting \(q\rightarrow1\) in \(J^{(2)}_{\nu}(x;q)\). Similarly to (4.1), we defined the second normalized q-Bessel function \(\mathcal{J}_{2}(x;a;q)\) by
$$\begin{aligned} \mathcal{J}_{2}(x;a;q)= {}_{0}\Phi_{1} \biggl( \begin{matrix} {0}\\ {qa} \end{matrix} \Big| q; -qax \biggr). \end{aligned}$$
(5.11)
There is a well-known limit from q-Laguerre [22] to the second normalized q-Bessel function as \(n \rightarrow \infty \)
$$ \mathcal{J}_{2}(x;a;q)=\lim_{n \rightarrow \infty }L_{n}(x,a;q). $$
(5.12)
From (5.12) and (5.3) is not difficult to establish the q-difference equation for \(\mathcal{J}_{2}(\lambda x,a;q)\)
$$\begin{aligned} (Y_{a,q,2}y) (x)=-a\lambda y(x), \end{aligned}$$
(5.13)
where
$$ (Y_{a,q,2}y) (x)=\frac{aq}{x}y(x)-q\frac{a+1}{x}y \bigl(q^{-1}x\bigr)+\frac {q}{x}y\bigl(q^{-2}x \bigr). $$
(5.14)
Furthermore, the q-Bessel operator \(Y_{a,q,2}\) is related to the q-Laguerre operator \(L_{a,q}\) defined in (5.4) by
$$ (L_{a,q}+a)f\bigl(q^{-1}x\bigr)=(Y_{a,q,2}+a)f( x). $$
(5.15)
This allows us to construct a Askey-Wilson algebra type that has the q-Bessel operator \(Y_{a,q,2}\) as one of its three generators. A straightforward computation shows that the operators X, Y, Z given by
$$\begin{aligned} &(Xf) (x):=xf(x), \\ & (Yf) (x):=\frac{q}{q^{2}-1}(Y_{a,q,2}+a)f(x),\\ & (Zf) (x):=-f\bigl(q^{-1}x\bigr), \end{aligned}$$
(5.16)
satisfy the relations
$$\begin{aligned} &YX -q^{2}XY = \mu_{1}Z +\mu_{2}X+ \mu_{3},\qquad ZY -qYZ = \mu_{4}Z,\qquad XZ -qZX =0, \end{aligned}$$
(5.17)
where
$$ \mu_{1}=-q\frac{1+a}{(1+q)}, \qquad\mu_{2}=-aq,\qquad \mu_{3}=-aq^{2},\qquad \mu_{4}=-\frac{aq}{(1+q)}. $$
(5.18)

Declarations

Acknowledgements

The third author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this Research group No. (RG-1435-026). The work of the second author is supported by King Saud University, Riyadh, DSFP, through the grant MATH 01.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, College of Sciences, King Saud University, Riyadh, Saudi Arabia
(2)
Department of Statistics & OR, College of Sciences, King Saud University, Riyadh, Saudi Arabia

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© Bouzeffour et al. 2015