Nagoya Mathematical Journal

$q$-Titchmarsh-Weyl theory: Series expansion

M. H. Annaby, Z. S. Mansour, and I. A. Soliman

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We establish a $q$-Titchmarsh-Weyl theory for singular $q$-Sturm-Liouville problems. We define $q$-limit-point and $q$-limit circle singularities, and we give sufficient conditions which guarantee that the singular point is in a limit-point case. The resolvent is constructed in terms of Green’s function of the problem. We derive the eigenfunction expansion in its series form. A detailed worked example involving Jackson $q$-Bessel functions is given. This example leads to the completeness of a wide class of $q$-cylindrical functions.

Article information

Nagoya Math. J. Volume 205 (2012), 67-118.

First available in Project Euclid: 1 March 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B24: Sturm-Liouville theory [See also 34Lxx] 34L10: Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions 39A13: Difference equations, scaling ($q$-differences) [See also 33Dxx]


Annaby, M. H.; Mansour, Z. S.; Soliman, I. A. q -Titchmarsh-Weyl theory: Series expansion. Nagoya Math. J. 205 (2012), 67--118. doi:10.1215/00277630-1543787.

Export citation


  • [1] L. D. Abreu, A q-sampling theorem related to the q-Hankel transform, Proc. Amer. Math. Soc. 133 (2004), 1197–1203.
  • [2] L. D. Abreu, Functions q-orthogonal with respect to their own zeros, Proc. Amer. Math. Soc. 134 (2006), 2695–2701.
  • [3] L. D. Abreu and J. Bustoz, “On the completeness of sets of q-Bessel functions” in Theory and Applications of Special Functions, Dev. Math. 13, Springer, New York, 2005, 29–38.
  • [4] L. D. Abreu, J. Bustoz, and J. L. Caradoso, The roots of the third Jackson q-Bessel functions, Int. J. Math. Math. Sci. 67 (2003), 4241–4248.
  • [5] M. H. Abu-Risha, M. H. Annaby, M. E. H. Ismail, and Z. S. Mansour, Linear q-difference equations, Z. Anal. Anwend. 26 (2007), 481–494.
  • [6] L. Ahlfors, Complex Analysis, Cambridge University Press, Cambridge, 1999.
  • [7] G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
  • [8] M. H. Annaby, q-Type sampling theorems, Results Math. 44 (2003), 214–225.
  • [9] M. H. Annaby and Z. S. Mansour, Basic Sturm-Liouville problems, J. Phys. A 38 (2005), 3775–3797; Correction, J. Phys. A 39 (2006), 8747.
  • [10] M. H. Annaby and Z. S. Mansour, On the zeros of basic finite Hankel transforms, J. Math. Anal. Appl. 323 (2006), 1091–1103.
  • [11] M. H. Annaby and Z. S. Mansour, On the zeros of the second and third Jackson q-Bessel functions and their associated q-Hankel transforms, Math. Proc. Cambridge Philos. Soc. 147 (2009), 47–67.
  • [12] M. H. Annaby and Z. S. Mansour, Asymptotic formulae for eigenvalues and eigenfunctions of q-Sturm-Liouville problems, Math. Nachr. 284 (2011), 443–470.
  • [13] J. M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, Amer. Math. Soc., Providence, 1968.
  • [14] B. M. Brown, J. S. Christiansen, and K. M. Schmidt, Spectral properties of a q-Sturm-Liouville operator, Comm. Math. Phys. 287 (2009), 259–274.
  • [15] B. M. Brown, W. D. Evans, and M. E. H. Ismail, The Askey-Wilson polynomials and q-Sturm-Liouville problems, Math. Proc. Cambridge Philos. Soc. 119 (1996), 1–16.
  • [16] J. Bustoz and J. L. Cardoso, Basic analog of Fourier series on a q-linear grid, J. Approx. Theory 112 (2001), 134–157.
  • [17] J. Bustoz and S. K. Suslov, Basic analog of Fourier series on a q-quadratic grid, Methods Appl. Anal. 5 (1998), 1–38.
  • [18] R. D. Carmichael, The general theory of linear q-difference equations, Amer. J. Math. 34 (1912), 147–168.
  • [19] R. D. Carmichael, Linear difference equations and their analytic solutions, Trans. Amer. Math. Soc. 12, no. 1 (1911), 99–134.
  • [20] R. D. Carmichael, On the theory of linear difference equations, Amer. J. Math. 35 (1913), 163–182.
  • [21] J. S. Christiansen and M. E. H. Ismail, A moment problem and a family of integral evaluations, Trans. Amer. Math. Soc. 358, no. 9 (2006), 4071–4097.
  • [22] R. Conway, Functions in One Complex Variable, Springer, New York, 1999.
  • [23] C. T. Fulton, Parametrizations of Titchmarsh’s m(λ)-functions in the limit circle case, Trans. Amer. Math. Soc. 229 (1977), 51–63.
  • [24] W. Hahn, Beiträge zur Theorie der Heineschen Reihen, Math. Nachr. 2 (1949), 340–379.
  • [25] M. Hajmirzaahmad and A. M. Krall, Singular second-order operators: The maximal and minimal operators, and selfadjoint operators in between, SIAM Rev. 34 (1992), 614–634.
  • [26] M. E. H. Ismail, The zeros of basic Bessel functions, the functions Jν+ax(x) and associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), 1–19.
  • [27] M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia Math. Appl. 98, Cambridge University Press, Cambridge, 2005.
  • [28] M. E. H. Ismail, On Jackson’s third q-Bessel function, preprint, 1996.
  • [29] F. H. Jackson, The applications of basic numbers to Bessel’s and Legendre’s equations, Proc. Lond. Math. Soc. (3) 2 (1905), 192–220.
  • [30] F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193–203.
  • [31] H. T. Koelink and R. F. Swarttouw, On the zeros of the Hahn-Exton q-Bessel function and associated q-Lommel polynomials, J. Math. Anal. Appl. 186 (1994), 690–710.
  • [32] T. H. Koornwinder and R. F. Swarttouw, On a q-analog of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. 333 (1992), no. 1, 445–461.
  • [33] S. Lang, Complex Analysis, Springer, New York, 1992.
  • [34] A. Lavagno, Basic-deformed quantum mechanics, Rep. Math. Phys. 64 (2009), 79–91.
  • [35] N. Levinson, A simplified proof of the expansion theorem for singular second order linear differential equations, Duke Math. J. 18 (1951), 57–71.
  • [36] B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, Amer. Math. Soc., Providence, 1975.
  • [37] B. M. Levitan and I. S. Sargsjan, Sturm Liouville and Dirac Operators, Kluwer, Dordrecht, 1991.
  • [38] A. Matsuo, Jackson integrals of Jordan-Pochhammer type and quantum Knizhnik-Zamolodchikov equations, Comm. Math. Phys. 151 (1993), 263–273.
  • [39] M. A. Naimark, Linear Differential Operators, Part II: Lineal Differential Operators in Hilbert Space, Frederick Ungar, New York, 1968.
  • [40] M. H. Stone, A comparison of the series of Fourier and Birkhoff, Trans. Amer. Math. Soc. 28 (1926), 695–761.
  • [41] M. H. Stone, Linear Transformations in Hilbert Space and Their Application to Analysis, Amer. Math. Soc., Providence, 1932.
  • [42] S. K. Suslov, Some expansions in basic Fourier series and related topics, J. Approx. Theory 115 (2002), 289–353.
  • [43] S. K. Suslov, An Introduction to Basic Fourier Series, Kluwer Ser. Dev. Math. 9, Kluwer Academic, Dordrecht, 2003.
  • [44] R. F. Swarttouw, The Hahn-Exton q-Bessel function, Ph.D. dissertation, Technical University of Delft, Delft, Netherlands, 1992.
  • [45] R. F. Swarttouw and H. G. Meijer, A q-analogue of the Wronskian and a second solution of the Hahn-Exton q-Bessel difference equation, Proc. Amer. Math. Soc. 120 (1994), 855–864.
  • [46] E. C. Titchmarsh, On the uniqueness of the Green’s function associated with a second-order differential equation, Canad. J. Math. 1 (1949), 191–198.
  • [47] E. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, I, 2nd ed., Clarendon Press, Oxford, 1962.
  • [48] H. Weyl, Gewöhnliche linear differentialgleichungen mit singularitäten stellen und ihre eigenfunktionen, Göttingen Ges. Wiss. Nach. 68 (1909), 73–64.
  • [49] H. Weyl, Gewöhnliche linear differentialgleichungen mit singularitäten stellen und ihre eigenfunktionen, Göttingen Ges. Wiss. Nach. 68 (1910), 442–467.
  • [50] H. Weyl, Über Gewöhnliche differentialgleichungen mit singularitäten und die zugehörigen entwicklung willkürlicher funktionen, Math. Ann. 68 (1910), 220–269.
  • [51] K. Yosida, On Titchmarsh-Kodaira’s formula concerning Weyl-Stone’s eingenfunction expansion, Nagoya Math. J. 1 (1950), 49–58.
  • [52] K. Yosida, Lectures on Differential and Integral Equations, Springer, New York, 1960.