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On the zeros of the second and third Jackson q-Bessel functions and their associated q-Hankel transforms

Published online by Cambridge University Press:  01 July 2009

M. H. ANNABY  and
Z. S. MANSOUR
M. H. ANNABY
Affiliation:
Department of Mathematics & Physics, Qatar University, P.O. Box 2713 Doha, Qatar. e-mail: [email protected], [email protected]
Z. S. MANSOUR
Affiliation:
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt. e-mail: [email protected]
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Abstract

We investigate the zeros of q-Bessel functions of the second and third types as well as those of the associated finite q-Hankel transforms. We derive asymptotic relations of the zeros of the q-Bessel functions by comparison with zeros of the theta function. The asymptotics of q-Bessel functions are also given. Zeros of finite q-Hankel transforms of q-summable functions are shown to be real and simple except for a finite number of possible non real zeros. Sufficient conditions are given to guarantee that all zeros are real. We give some applications concerning zeros of combinations of q-Bessel functions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 1 , July 2009 , pp. 47 - 67
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]Abreu, L. D., Bustoz, J. and Caradoso, J. L.The roots of the third Jackson q-Bessel functions. Internat. J. Math. Math. Sci. 67 (2003), 4241–4248.CrossRefGoogle Scholar
[2]Annaby, M. H. and Mansour, Z. S.A basic analog of a theorem of pólya. Math. Z. 258 (2008), 363–379.CrossRefGoogle Scholar
[3]Annaby, M. H. and Mansour, Z. S.On the zeros of basic finite Hankel transforms. J. Math. Anal. Appl. 323 (2006), 1091–1103.CrossRefGoogle Scholar
[4]Annaby, M. H., Mansour, Z. S. and Ashour, O. A. On the reality and asymptotics of zeros of finite q-Hankel transforms. J. Approx. Theory (2008), doi:10.1016/j.jat.2008.11.001.CrossRefGoogle Scholar
[5]Bergweiler, W. and Hayman, W. K.Zeros of solutions of a functional equation. Comp. Meth. Func. Theory. 3 (2003), 55–78.CrossRefGoogle Scholar
[6]Boas, R. P.Entire Functions (Academic Press, 1954).Google Scholar
[7]Bustoz, J. and Cardoso, J. L.Basic analog of Fourier series on a q-linear grid. J. Approx. Theory. 112 (2001), 154–157.CrossRefGoogle Scholar
[8]Chen, Y., Ismail, M. E. H. and Muttalib, K. A.Asymptotics of basic Bessel functions q-Laguerre polynomials. J. Comput. Appl. Math. 54 (1994), 263–272.CrossRefGoogle Scholar
[9]Olde Daalhuis, A. B.Asymptotic expansions for q-gamma, q-exponential and q-Bessel functions. J. Math. Anal. Appl. 186 (1994), 896–913.CrossRefGoogle Scholar
[10]Gasper, G. and Rahman, M.Basic Hypergeometric Series (Cambridge University Press 2004).CrossRefGoogle Scholar
[11]Hayman, W. K.Subharmonic Functions, II (Academic Press, 1989).Google Scholar
[12]Hayman, W. K.On the zeros of q-bessel function. Contemp. Math. 382 (2005), 205–216.CrossRefGoogle Scholar
[13]Ismail, M. E. H.The zeros of basic Bessel functions, the functions J ν+ax(x) and associated orthogonal polynomials. J. Math. Anal. Appl. 86 (1982), 11–19.CrossRefGoogle Scholar
[14]Ismail, M. E. HClassical and Quantum Orthogonal Polynomials in One Variable. (Cambridge University Press, 2005).CrossRefGoogle Scholar
[15]Ismail, M. E. H. and Zhang, C.Zeros of entire functions and a problem of Ramanujan. Adv. Math. 209 (2007), 363–380.CrossRefGoogle Scholar
[16]Jackson, F. H.A basic-sine and cosine with sympolical solutions of certain differential equations. proc. Edinburgh Math. Soc. 22 (1903–1904), 28–34.CrossRefGoogle Scholar
[17]Jackson, F. H.The applications of basic numbers to Bessel's and Legendre's equations. Proc. Lond. Math. Soc. (2). 2 (1905), 192–220.CrossRefGoogle Scholar
[18]Jackson, F. H.The basic gamma function and elliptic functions. Proc. Roy. Soc. A. 76 (1905), 127–144.Google Scholar
[19]Jackson, F. H.On q-definite integrals. Quart. J. Pure and Appl. Math. 41 (1910), 193–203.Google Scholar
[20]Katkova, O. M. and Vishnyakova, A. M.A sufficient condition for a polynomial to be stable. J. Math. Anal. Appl. 347 (2008), 81–89.CrossRefGoogle Scholar
[21]Koelink, H. T. and Swarttouw, R. F.On the zeros of the Hahn–Exton q-Bessel function and associated q−Lommel polynomials. J. Math. Anal. Appl. 186 (1994), 690–710.CrossRefGoogle Scholar
[22]Koornwinder, T. H. and Swarttouw, R. F.On a q−analog of Fourier and Hankel transforms. Trans. Amer. Math. Soc. 333 (1992), 445–461.Google Scholar
[23]Kvitsinsky, A. A.Zeta functions, heat kernel expansions, and asymptotics for q-Bessel functions. J. Math. Anal. Appl. 196 (1995), 947–964.CrossRefGoogle Scholar
[24]Levin, B. Ja.Distribution of Zeros of Entire Functions. Translations of Mathematical Monographs 5 (A.M.S. Providence, 1980).Google Scholar
[25]Pólya, G. and Szegő, G.Über die Nullstellen gewisser ganzer Funktionen. Math. Z. 2 (1918), 352–383.CrossRefGoogle Scholar
[26]Pólya, G. and Szegő, G.Problems and Theorems in Analysis I. (Springer-Verlag, 1972).Google Scholar