On $q$-analogues of the Fourier and Hankel transforms
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- by Tom H. Koornwinder and René F. Swarttouw
- Trans. Amer. Math. Soc. 333 (1992), 445-461
- DOI: https://doi.org/10.1090/S0002-9947-1992-1069750-0
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Abstract:
For H. Exton’s $q$-analogue of the Bessel function (going back to W. Hahn in a special case, but different from F. H. Jackson’s $q$-Bessel functions) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to orthogonality relations which are $q$-analogues of the Hankel integral transform pair. These results are implicit, in the context of quantum groups, in a paper by Vaksman and Korogodskiĭ. As a specialization we get ($q$-cosines and $q$-sines which admit $q$-analogues of the Fourier-cosine and Fourier-sine transforms. We also get a formula which is both an analogue of Graf’s addition formula and of the Weber-Schafheitlin discontinuous integral.References
- George E. Andrews, $q$-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra, CBMS Regional Conference Series in Mathematics, vol. 66, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 858826, DOI 10.1090/cbms/066
- George E. Andrews and Richard Askey, Enumeration of partitions: the role of Eulerian series and $q$-orthogonal polynomials, Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976) NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol. 31, Reidel, Dordrecht-Boston, Mass., 1977, pp. 3–26. MR 519776
- Harold Exton, A basic analogue of the Bessel-Clifford equation, Jñānābha 8 (1978), 49–56. MR 637273 —, $q$-hypergeometric functions and applications, Ellis Horwood, 1983.
- Philip Feinsilver, Elements of $q$-harmonic analysis, J. Math. Anal. Appl. 141 (1989), no. 2, 509–526. MR 1009060, DOI 10.1016/0022-247X(89)90194-7
- George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153
- Wolfgang Hahn, Beiträge zur Theorie der Heineschen Reihen. Die $24$ Integrale der Hypergeometrischen $q$-Differenzengleichung. Das $q$-Analogon der Laplace-Transformation, Math. Nachr. 2 (1949), 340–379 (German). MR 35344, DOI 10.1002/mana.19490020604
- Wolfgang Hahn, Die mechanische Deutung einer geometrischen Differenzgleichung, Z. Angew. Math. Mech. 33 (1953), 270–272 (German). MR 57662, DOI 10.1002/zamm.19530330811 E. Heine, Handbuch der Kugelfunctionen, Theorie und Anwendungen. Erster Band: Theorie der Kugelfunctionen und der verwandten Functionen, Reiner, Berlin, 1878.
- Mourad E. H. Ismail, The zeros of basic Bessel functions, the functions $J_{\nu +ax}(x)$, and associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), no. 1, 1–19. MR 649849, DOI 10.1016/0022-247X(82)90248-7
- Tom H. Koornwinder, Jacobi functions as limit cases of $q$-ultraspherical polynomials, J. Math. Anal. Appl. 148 (1990), no. 1, 44–54. MR 1052043, DOI 10.1016/0022-247X(90)90026-C
- Mizan Rahman, An addition theorem and some product formulas for $q$-Bessel functions, Canad. J. Math. 40 (1988), no. 5, 1203–1221. MR 973517, DOI 10.4153/CJM-1988-051-7 —, A note on the orthogonality of Jackson’s $q$-Bessel functions, Canad. Math. Bull. 32 (1989), 369-376.
- L. L. Vaksman and L. I. Korogodskiĭ, The algebra of bounded functions on the quantum group of motions of the plane and $q$-analogues of Bessel functions, Dokl. Akad. Nauk SSSR 304 (1989), no. 5, 1036–1040 (Russian); English transl., Soviet Math. Dokl. 39 (1989), no. 1, 173–177. MR 997179
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 333 (1992), 445-461
- MSC: Primary 33D45; Secondary 33D80, 39A10, 44A15
- DOI: https://doi.org/10.1090/S0002-9947-1992-1069750-0
- MathSciNet review: 1069750