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

The aim of this work is to study new functions arising from the limit transition of the Jackson's $q$-Bessel functions when $q\rightarrow -1$. These functions coincide with the $cas$ function for particular values of their parameters. We prove also that these functions are eigenfunction of differential-difference operators of Dunkl-type. Further, we consider special cases of the Askey-Wilson algebra $AW(3)$ that have these operators (up to constants) as one of their three generators and whose defining relations are given in terms of anticommutators.


Introduction
Most of the special functions and orthogonal polynomials in the Askey-Wilson scheme are limit cases of the Askey-Wilson polynomials, which are at the top level with five parameters (see [15,16]). The Askey-Wilson algebra AW (3) (see [28]) describes deeper symmetries of the Askey-Wilson polynomials and is related to the quantum groups U q (sl 2 ), U q (su 2 ) (see [17]) and the double affine Hecke algebra of type (C ∨ 1 , C 1 ) (see [18,19]). Limits of basic hypergeometric orthogonal polynomials as q → 1 are well-known (see [15,Ch.5,Ch.4], [16]). Often, the q → 1 limit of the polynomials corresponds to the limit q → 1 of the algebra AW (3) to classical algebras. In this paper we are concerned with limits for q → −1 of some q-special functions and orthogonal polynomials in the Askey-Wilson schema [15,16] and their related Askey-Wilson algebras type. Recently, L. Vinet and A. Zhedanov [25] introduced new explicit families of orthogonal polynomials by considering appropriate limits q → −1 of the big and little q-Jacobi polynomials. They show that the big and little (−1)-Jacobi polynomials are eigenfunctions of first order Dunkl-type operators which are a mixture of a differential and a real reflection operator s, which acts on a function f (x) as (sf )(x) = f (−x). The limits q → −1 of q-special functions in the Askey-Wilson functions scheme (see [14]) are not yet studied and here we will discuss the Jackson's q-Bessel functions (see [11,12,13]) and q-Laguerre cases (see [7, §18.27(v)]), that we define next. The Jackson q-Bessel functions are introduced first by F. H. Jackson. The first and the second q-Bessel are intensively studied by Ismail [10]. The third q-Bessel is rediscovered later by W. Hahn [8] in a special case and by H. Exton [5]. The third q-Bessel function has also an interpretation as matrix elements of irreducible representations of the quantum group of plane motions E q (2) [23] and satisfies an orthogonality relations that it makes more suitable for harmonic analysis (see [20,1]). Of course, when q → 1 the Jackson's q-Bessel functions tend to the standard Bessel function. In this paper we show that the limit q → −1 of the third and second q-Bessel functions lead to a new type of nonsymmetric Bessel function, which is an eigenfunction of first order differential-difference operator related to the Dunkl operator associated to a root system of type A 1 and coincides for a particular value of its parameter with the cas function [9]. Furthermore, there is a limit transition from little q-Jacobi polynomials to the third q-Bessel function and from q-Laguerre to the second q-Bessel function, which allows us to construct a q-Bessel version of the Askey-Wilson AW (3) algebra. Notations Throughout assume −1 < q < 1. For q-Pochhammer symbols and q-hypergeometric series use the notation of [6].
There is a q-difference equation of the form The Askey-Wilson algebra AW (3) was introduced by Zhedanov in 1991 (see [28]). The algebra AW (3) involves a nonzero scalar q and three parameters ω 1 , ω 2 and ω 3 . Given these data, the corresponding algebra AW (3) is defined as an associative algebra, generated by the elements X, Y and Z, subject to the commutation relations There is a central element Q which is explicitly given as a polynomial of degree 3 in the generators In particular, in the case of little q-Jacobi polynomials, the related Askey-Wilson algebra is generated by X, Y , Z with relations [25] The constants ω 1 , ω 2 and ω 3 (together with the value of the Casimir operator Q) define representations of the AW (3) algebra (see [28] for details). There is a representation (the basic representation) of the little q-Jacobi AW (3) algebra with structure constants (2.8) on the space of polynomials as follows: The Casimir operator have been introduced and investigated in [25] as limits of the little q-Jacobi polynomials (2.1), when q → −1. More precisely We recall here their basic properties [25]. The polynomial P (α,β,−1) n (x) satisfies the difference-differential equation and λ n = −n, if n is even, α + β + n + 1, if n is odd . These polynomials have the following expressions in terms of the hypergeometric series: For n, even and for n odd Introduce the operators In [25], it was shown that these operators are closed in frame of the Askey-Wilson algebra and that they satisfy the commutation relations The Casimir operator is equal to takes the value Q = 1.

The nonsymmetric Hankel transform
For Bessel functions J α (x), see [7, Ch.10] and references given therein. Let us consider the normalized Bessel function J α (x), which is given by is an entire function and we have the simple properties and special cases (see [ The function x → J α (λx) satisfies the eigenvalue equation [7, §10.13.5]: The Hankel transform pair [7, §10. 22(v)], for f in a suitable function class, is given by . Now consider the so-called nonsymmetric Bessel function, also called Dunkl-type Bessel function, in the rank one case (see [3, §4]): In particular, The nonsymmetric Hankel transform pair takes the form . The transform pair (3.4) follows immediately from (3.2). For given α define the differentialreflection operator (3.5) ( This is the Dunkl operator for root system A 1 (see [4,Definition 4.4.2)]). Then we have the eigenvalue equation 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 [7, (10.6.2)]): The double degeneration of the double affine Hecke algebra H" is generated by D, Z and s with relations The operator D, s and Z acts on the polynomial f (x) as: The operators defined by (3.8) are also known as papa-Bose operators and the algebra (3.7) is equivalent to the para-Bose algebra [21,22]. Another important generalization of the exponential function is the so-called cas function, which is given by [9,2] (3.9) cas(x) = cos(x) + sin(x).
It is clearly that the function y(x) = cas(λx) is unique C ∞ -solution of the the differentialdifference problem where Λ = s∂ and ∂ is the derivative operator.

Limit q → −1 of the third q-Bessel function
Third q-Bessel function was introduced in [8,5], see also [20,1,5], which is defined by We will consider a slightly different function J 3 (x, a; q), called normalized third q-Bessel function, which is defined by It is easy to see that lim The function J 2 (λx, a; q) is a solution of the q-difference Next, we describe the construction of new function by the limiting process from normalized third q-Bessel as q → −1 and from the (−1)-Jacobi polynomials as n → ∞. This function coincides with the cas function for a particular value of its parameter. Moreover, we have the following diagram for limit relations between these special functions and orthogonal polynomials.
little q-Jacobi There is a well-known limit from Jacobi polynomials to Bessel functions, see [7, (18.11.5)], The q-analogue of this limit transition starts with the little q-Jacobi polynomials (2.1). From Proposition A.1 in [20], we have The operators X, Y and Z defined in (2.9) have also a limit for n → ∞ after the rescaling x → q n x. More precisely, let us denote by , Then in the limit the operators X, Y and Z are given by When n → ∞, the relations (2.13) become where (4.8) The resulting algebra generated by X, Y and Z with relations (4.7) and (4.8) is called the q-Bessel AW (3) algebra. In this case the Casimir operator becomes and takes the value Q = −a.
The relations (4.7) hold in the limit q → −1. Indeed, let us take the parametrization a = −e ε(2α+1) and q = −e ε . Since, the q-difference equation (4.2) tends formally as ε → 0 to the differential-difference equation The operator Y α is a difference-differential operator of the first order containing reflection terms. Notice the important property of the operator Y α : 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 α . Now, let us introduce the operators Then it is elementary to verify that the operators X, Y , Z satisfy the relations which corresponds to the AW (3) algebra with parameters It is easily verified that the Casimir operator is (4.14) In the case of the realization (4.12) of the operators X, Y Z, the Casimir operator becomes the identity operator.

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 we have used slightly different notations (see [7, (18.27.15)]). They satisfy the recurrence relations There is a q-difference equation of the form When q → 1 (a = q α ) the q-Laguerre polynomial L n (x, a; q) becomes the ordinary Laguerre polynomial There is a limit transition from little q-Jacobi to q-Laguerre, (see [16, §4.12.2]) Starting with the operators X, Y and Z given by (2.9) we can also obtain the following operators Then where the operator L a,q is defined in (5.4). A simple computation shows that the operators X, Y , Z satisfy the relations Y X − qXY = Z + ω 3 , ZY − qY Z = ω 2 , XZ − qZX = 0, (5.8) where (5.9) The Casimir operator (5.10) Q = (q 2 − 1)Y XZ + Z 2 + (q + 1)(ω 2 qX + ω 3 Z) takes the value Q = −aq 2 .

Second
Jackson's q-Bessel function. The second Jackson's q-Bessel function is defined as follows This notation is from [10] and is deferent from Jackson's notation (see [11,12,13]). The classical Bessel Function J ν is recovered by letting q → 1, in J ν (x; q). Similarly to (4.1), we defined the second normalized q-Bessel function J 2 (x; a; q) by J 2 (x; a; q) = 0 Φ 1 0 qa q; −qax . There is a well-known limit from q-Laguerre [16] to the second normalized q-Bessel function as n → ∞ (5.12) J 2 (x; a; q) = lim n→∞ L n (x, a; q).
Furthermore, the q-Bessel operator Y a,q,2 is related to the q-Laguerre operator L a,q defined in (5.4) by (5.15) (L a,q + a)f (q −1 x) = (Y a,q,2 + a)f (x).
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 .