- Research
- Open Access
Jackson’s \((-1)\)-Bessel functions with the Askey-Wilson algebra setting
https://doi.org/10.1186/s13662-015-0612-6
© Bouzeffour et al. 2015
- 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 functions
- difference-differential equations
- Askey-Wilson algebra
MSC
- 33D45
- 33D80
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 [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
2 Askey-Wilson relations for the little q-Jacobi polynomials
2.1 Little q-Jacobi polynomials
2.2 Little \((-1)\)-Jacobi polynomials
3 The nonsymmetric Hankel transform
4 Limit \(q\rightarrow-1\) of the third q-Bessel function
The upper limit in the previous diagram is studied by [1]. Next we are concerned only by the three other limits.
Theorem 4.1
Proof
5 The second q-Bessel function case
5.1 q-Laguerre polynomials
5.2 Second Jackson’s q-Bessel function
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
References
- Vinet, L, Zhedanov, A: A ‘missing’ family of classical orthogonal polynomials. J. Phys. A, Math. Gen. 44, 085201 (2011) MathSciNetView ArticleGoogle Scholar
- Jackson, FH: On generalized functions of Legendre and Bessel. Trans. R. Soc. Edinb. 41, 1-28 (1903) Google Scholar
- Jackson, FH: Theorems relating to a generalization of the Bessel functions. Trans. R. Soc. Edinb. 41, 105-118 (1903) View ArticleGoogle Scholar
- Jackson, FH: The application of basic numbers to Bessel’s and Legendre’s functions. Proc. Lond. Math. Soc. 2, 192-220 (1903/1904) Google Scholar
- Ismail, MEH: The zeros of basic Bessel functions, the functions \(J_{\nu+ax}\) and associated orthogonal polynomials. J. Math. Anal. Appl. 86, 1-19 (1982) MathSciNetView ArticleMATHGoogle Scholar
- Hahn, W: Beiträge zur Theorie der Heineschen Reihen. Math. Nachr. 2, 340-379 (1949) MathSciNetView ArticleMATHGoogle Scholar
- Exton, H: A basic analogue of the Bessel-Clifford equation. Jñānābha 8, 49-56 (1978) MathSciNetMATHGoogle Scholar
- Vaksman, LL, Korogodski, LI: An algebra of bounded functions on the quantum group of the motions of the plane and q-analogues of Bessel functions. Sov. Math. Dokl. 9, 173-177 (1989) Google Scholar
- Koornwinder, TH, Swarttouw, RF: On q-analogues of the Fourier and Hankel transforms. Trans. Am. Math. Soc. 333, 445-461 (1992) MathSciNetMATHGoogle Scholar
- Bettaibi, N, Bouzeffour, F, Elmonser, HB, Binous, W: Elements of harmonic analysis related to the third basic zero order Bessel function. J. Math. Anal. Appl. 342, 1203-1219 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Watson, GN: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944) MATHGoogle Scholar
- Hartley, RVL: A more symmetrical Fourier analysis applied to transmission problems. Proc. IRE 30, 144-150 (1942) MathSciNetView ArticleMATHGoogle Scholar
- Gasper, G, Rahman, M: Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004) View ArticleMATHGoogle Scholar
- Olver, FWJ, Lozier, DW, Boisvert, RF, Clark, CW (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010) http://dlmf.nist.gov MATHGoogle Scholar
- Zhedanov, AS: ‘Hidden symmetry’ of Askey-Wilson polynomials. Teor. Mat. Fiz. 89, 190-204 (1991); (English transl.: Theor. Math. Phys. 89, 1146-157 (1991)) MathSciNetView ArticleMATHGoogle Scholar
- Dunkl, CF: Hankel transforms associated to finite reflection groups. In: Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications. Contemp. Math., vol. 138, pp. 123-138. Am. Math. Soc., Providence (1992) View ArticleGoogle Scholar
- Dunkl, CF, Xu, Y: Orthogonal Polynomials of Several Variables. Cambridge University Press, Cambridge (2001) View ArticleMATHGoogle Scholar
- Cherednik, I, Markov, Y: Hankel transform via double Hecke algebra. In: Iwahori-Hecke Algebras and Their Representation Theory. Lecture Notes in Math., vol. 1804, pp. 1-25. Springer, Berlin (2002) View ArticleGoogle Scholar
- Ohnuki, Y, Kamefuchi, S: Quantum Field Theory and Parastatistics. Springer, Berlin (1982) View ArticleMATHGoogle Scholar
- Rosenblum, M: Generalized Hermite polynomials and the Bose-like oscillator calculus. In: Nonselfadjoint Operators and Related Topics. Operator Theory: Advances and Applications, vol. 73, pp. 369-396 (1994) arXiv:math/9307224 View ArticleGoogle Scholar
- Bracewell, RN: The Hartley Transform. Oxford University Press, New York (1986) MATHGoogle Scholar
- Koekoek, R, Swarttouw, RF: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology (1998) http://aw.twi.tudelft.nl/~koekoek/askey/