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

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.

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 [2–4]. 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 [8–10]. 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

$$ (a_{1}, \ldots, a_{r} ; q )_{n}:= (a_{1}; q )_{n}\cdots (a_{r}; q )_{n}. $$

Askey-Wilson relations for the little q-Jacobi polynomials

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 $Q=-b^{-1}$.

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

$$Q=1. $$

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.

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:

$$\begin{aligned} &\lim_{\varepsilon\rightarrow0}\varepsilon^{-[n/2]}\bigl(-e^{\varepsilon \alpha}; -e^{\varepsilon}\bigr)_{n}=(-1)^{[n/2]}2^{n}\bigl(( \alpha+1)/2\bigr)_{[n/2]},\\ & \lim_{\varepsilon\rightarrow0}\varepsilon ^{-[(n+1)/2]}\bigl(e^{\varepsilon \alpha} ;-e^{\varepsilon}\bigr)_{n}=(-1)^{([n+1)/2]}2^{n}( \alpha/2)_{[(n+1)/2]}. \end{aligned}$$

The second q-Bessel function case

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 $Q=-aq^{2}$.

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 [2–4]. 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)