Abstract
In an attempt to find a q-analogue of Weber and Schafheitlin's integral ∫∞ 0 x −ρ J μ (ax) J ν (bx) dx which is discontinuous on the diagonal a = b the integral ∫∞ 0 x −ρ J (2) ν (a(1 − q)x; q)J (1) μ (b(1 − q)x; q) dx is evaluated where J (1) μ (x; q) and J (2) μ (x; q) are two of Jackson's three q-Bessel functions. It is found that the question of discontinuity becomes irrelevant in this case. Evaluations of this integral are also made in some interesting special cases. A biorthogonality formula is found as well as a Neumann series expansion for x ρ in terms of J (2) ν+1+2n ((1 − q)x; q). Finally, a q-Lommel function is introduced.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Erdélyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi (ed.), Higher Transcendental Functions, Vol II, McGraw-Hill, New York, 1953.
H. Exton, “A basic analogue of the Bessel-Clifford equation,” Jñãnãbha 8 (1978) 49–56.
H. Exton, q-Hypergeometric Functions and Applications, Ellis Horwood, 1983.
G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, 1990.
W. Hahn, “Beiträge zur theorie der Heineschen Reichen,” Math. Nachr. 2 (1949) 340–379.
W. Hahn, “Die mechanische Deutung einer geometrischen Differenzgleichung,” Z. Angew. Math. Mech. 33 (1953) 270–272.
M.E.H. Ismail, “The basic Bessel functions and polynomials,” SIAM J. Math. Anal. 12 (1981) 454–468.
M.E.H. Ismail, “The zeros of basic Bessel functions, the functions J v+ax. (x) and the associated orthogonal polynomials,” J. Math. Anal. Appl. 86 (1982) 1–19.
M.E.H. Ismail, “Some properties of Jackson's third q-Bessel function,” to appear.
M.E.H. Ismail, D. Masson, and S.K. Suslov, “The q-Bessel functions on a quadratic grid,” Algebraic Methods and q-Special Functions (Montreal, 1996) 183-200, CRM Lecture Notes 22, Amer. Math. Soc., Providence, RI, 1999.
M.E.H. Ismail, D. Masson, and S.K. Suslov, “Properties of a q-analogue of Bessel functions,” to appear.
F.H. Jackson, “On generalized functions of Legendre and Bessel,” Trans. Roy. Soc. Edin. 41 (1904) 1–28.
F.H. Jackson, “Applications of basic numbers to Bessel's and Legendre's functions,” Proc. Lond. Math. Soc. 2(2) (1905) 192–220.
H.T. Koelink, “The quantum group of plane motions and basic Bessel functions,” Indag. Mathem., N.S. 6(2) (1995) 197–211.
T.H. Koornwinder and R.F. Swarttouw, “On q-analogues of the Fourier and Hankel transforms,” Trans. Amer. Math. Soc. 333 (1992) 445–461.
M. Rahman, “Some infinite integrals of q-Bessel functions,” Ramanujan International Symposium on Analysis, Proceedings of Ramanujan Birth Centenary Year: Pune, India, N. K. Thakore, ed. 1987, pp. 117–138.
M. Rahman, “An addition theorem and some product formulas for q-Bessel functions,” Can. J. Math. 40 (1988) 1203–1221.
S. Ramanujan, “Some definite integrals,” Mess. Math. 44 (1915) 10–18; reprinted in Collected papers of Srinivasa Ramanujan (G.H. Hardy, P.V. Seshu Aiyer, and B.M. Wilson, eds.), Cambridge University Press, 1927; reprinted by Chelsea, New York, 1962.
S.K. Suslov, “Some orthogonal very-well-poised 8 φ 7 functions that generalize Askey-Wilson polynomials,” MSRI Preprint#1997-060, Berkeley, California, 1997.
R.F. Swarttouw, “The Hahn-Exton q-Bessel Functions,” Ph.D. Thesis, Delft Technical University, 1992.
G.N. Watson, “Theory of Bessel Functions,” Cambridge University Press, 1944.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rahman, M. A q-Analogue of Weber-Schafheitlin Integral of Bessel Functions. The Ramanujan Journal 4, 251–265 (2000). https://doi.org/10.1023/A:1009892718531
Issue Date:
DOI: https://doi.org/10.1023/A:1009892718531