Skip to main content
SpringerLink
Log in

q-Sturm–Liouville Problems

  • Chapter
  • First Online:
Book cover q -Fractional Calculus and Equations

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2056))

  • 2431 Accesses

Abstract

In this chapter we introduce the study held by Annaby and Mansour in (J. Phys. A Math. Gen. 38(17), 3775–3797, 2005) of a self adjoint basic Sturm–Liouville eigenvalue problem in a Hilbert space. The last two sections of this chapter are about the q 2-Fourier transform introduced by Rubin in (J. Math. Anal. Appl. 212(2), 571–582, 1997; Proc. Am. Math. Soc. 135(3), 777–785, 2007), when q lies in a proper subset of (0, 1) and the generalization of Rubin’s q 2-Fourier transform, introduced in (Mansour, Generalizations of Rubin’s q 2-fourier transform and q-difference operator, submitted, 2012) for any q ∈ (0, 1).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. H.I. Abdel-Gawad, A.A. Aldailami, On q-dynamic equations modelling and complexity. Appl. Math. Model. 34(3), 697–709 (2010)

    Google Scholar 

  2. T. Abdeljawad, D. Baleanu, Caputo q-fractional initial value problems and a q-analogue Mittag-Leffler function. Comm. Nonlinear Sci. Numer. Simul. 16(12), 4682–4688 (2011)

    Google Scholar 

  3. W.H. Abdi, On q-Laplace transforms. Proc. Natl. Acad. Sci. India (Sect. A) 29, 389–408 (1960)

    Google Scholar 

  4. W.H. Abdi, Application of q-Laplace transform to the solution of certain q-integral equations. Rend. Circ. Mat. Palermo 11(2), 245–257 (1962)

    Google Scholar 

  5. W.H. Abdi, On certain q-difference equations and q-Laplace transforms. Proc. Natl. Acad. Sc. India A 28, 1–15 (1962)

    Google Scholar 

  6. W.H. Abdi, Certain inversion and representation formulae for q-Laplace transforms. Math. Z. 83, 238–249 (1964)

    Google Scholar 

  7. L.D. Abreu, A q-sampling theorem related to the q-Hankel transform. Proc. Am. Math. Soc. 133, 1197–1203 (2005)

    Google Scholar 

  8. L.D. Abreu, Sampling theory associated with q-difference equations of the Sturm-Liouville type. J. Phys. A 38(48), 10311–10319 (2005)

    Google Scholar 

  9. L.D. Abreu, J. Bustoz, J.L. Caradoso, The roots of the third Jackson q-Bessel functions. Int. J. Math. Math. Sci. 2003(67), 4241–4248 (2003)

    Google Scholar 

  10. M.H. Abu-Risha, M.H. Annaby, M.E.H. Ismail, Z.S. Mansour, Linear q-difference equations. Z. Anal. Anwend. 26(4), 481–494 (2007)

    Google Scholar 

  11. C.R. Adams, Note on the existence of analytic solutions of non-homogeneous linear q-difference equations, ordinary and partial. Ann. Math. 27(2), 73–83 (1925)

    Google Scholar 

  12. C.R. Adams, Note on the existence of analytic solutions of non-homogeneous linear q-difference equations, ordinary and partial. Ann. Math. (2) 30(1–4), 626 (1928/1929)

    Google Scholar 

  13. C.R. Adams, On the linear ordinary q-difference equation. Ann. Math. (2) 30(1–4), 195–205 (1928/1929)

    Google Scholar 

  14. M. Adivar, M. Bohner, Spectral analysis of q-difference equations with spectral singularities. Math. Comput. Modell. 43(7–8), 695–703 (2006)

    Google Scholar 

  15. R.P. Agarwal, Fractional q-derivative and q-integrals and certain hypergeometric transformations. Ganita 27(1–2), 25–32 (1976)

    Google Scholar 

  16. R.P. Agarwal, A propos d’une note de M. Pierre Humbert (French). C. R. Acad. Sci. Paris 236, 2031–2032 (1953)

    Google Scholar 

  17. R.P. Agarwal, Certain fractional q-integrals and q-derivatives. Proc. Camb. Phil. Soc. 66, 365–370 (1969)

    Google Scholar 

  18. W.A. Al-Salam, q-analogues of Cauchy’s formulas. Proc. Am. Math. Soc. 17, 616–621 (1966)

    Google Scholar 

  19. W.A. Al-Salam, Some fractional q-integrals and q-derivatives. Proc. Edinb. Math. Soc. 2(15), 135–140 (1966/1967)

    Google Scholar 

  20. W.A. Al-Salam, A. Verma, A fractional Leibniz q-formula. Pac. J. Math. 60(2), 1–9 (1975)

    Google Scholar 

  21. T.S. Aleroev, On the problem of the zeros of the Mittag-Leffler function and the spectrum of a fractional-order differential operator. Differ. Equat. 36(9), 1414–1415 (2000). Translation from Differ. Uravn. 36(9), 1278–1279 (2000)

    Google Scholar 

  22. T.S. Aleroev, H.T. Aleroeva, A problem on the zeros of the Mittag–Leffler function and the spectrum of a fractional-order differential operator. Electron. J. Qual. Theor. Differ. Equat. 2009(25), 18 (2009)

    Google Scholar 

  23. A. Algin, A comparative study on q-deformed fermion oscillators. Int. J. Theor. Phys. 50(5), 1554–1568 (2011)

    Google Scholar 

  24. G.E. Andrews, The Theory of Partitions, vol. 2 of Encyclopedia of Mathematics and Applications (Addison-Wesley, London, 1976)

    Google Scholar 

  25. G.E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge University Press, Cambridge, 1999)

    Google Scholar 

  26. M.H. Annaby, q-type sampling theorems. Result. Math. 44(3–4), 214–225 (2003)

    Google Scholar 

  27. M.H. Annaby, J. Bustoz, M.E.H. Ismail, On sampling theory and basic Sturm–Liouville systems. J. Comput. Appl. Math. 206(1), 73–85 (2007)

    Google Scholar 

  28. M.H. Annaby, P.L. Butzer, Sampling in Paley-Wiener spaces associated with fractional integro-differential operators. J. Comput. Appl. Math. 171(1–2), 39–57 (2004)

    Google Scholar 

  29. M.H. Annaby, H.A. Hassan, Z.S. Mansour, Sampling theorems associated with singular q-Sturm Liouville problems. doi:10.1007/s00025-011-0134-9

    Google Scholar 

  30. M.H. Annaby, Z.S. Mansour, Basic Sturm Liouville problems. J. Phys. A Math. Gen. 38(17), 3775–3797 (2005). Corrigendum, J. Phys. A Math. Gen. 39(27), 8747 (2006)

    Google Scholar 

  31. M.H. Annaby, Z.S. Mansour, On the zeros of basic finite Hankel transforms. J. Math. Anal. Appl. 323(2), 1091–1103 (2006)

    Google Scholar 

  32. M.H. Annaby, Z.S. Mansour, A basic analog of a theorem of Pólya. Math. Z. 258(2), 363–379 (2008)

    Google Scholar 

  33. M.H. Annaby, Z.S. Mansour, q-Taylor and interpolation series for Jackson q-difference operators. J. Math. Anal. Appl. 334(1), 472–483 (2008)

    Google Scholar 

  34. M.H. Annaby, Z.S. Mansour, On zeros of second and third Jackson q-Bessel functions and their q-integral transforms. Math. Proc. Camb. Philos. Soc. 147(1), 47–67 (2009)

    Google Scholar 

  35. M.H. Annaby, Z.S. Mansour, Asymptotic formulae for eigenvalues and eigenfunctions of q-Sturm-Liouville problems. Math. Nachr. 284(4), 443–470 (2011)

    Google Scholar 

  36. M.H. Annaby, Z.S. Mansour, O.A. Ashour, Sampling theorems associated with biorthogonal q-Bessel functions. J. Phys. A Math. Theor. 43(29), 15 (2010)

    Google Scholar 

  37. M.H. Annaby, Z.S. Mansour, I.A. Soliman, q − titchmarsh-weyl theory: Series expansion. Nagoya Math. J. 205, 67–118 (2012)

    Google Scholar 

  38. I. Area, E. Godoy, F. Marcellán, J.J. Moreno-Balcázar, Inner products involving q-differences: The little q-Laguerre-Sobolev polynomials. Higher transcendental functions and their applications. J.  Comput. Appl. Math. 118(1–2), 1–22 (2000)

    Google Scholar 

  39. I. Area, E. Godoy, A. Ronveaux, A. Zarzo, Hypergeometric type q-difference equations: Rodrigues type representation for the second kind solution. J. Comput. Appl. Math. 173(1), 81–92 (2005)

    Google Scholar 

  40. R. Askey, The q-gamma and q-beta functions. Appl. Anal. 8(2), 125–141 (1978/1979)

    Google Scholar 

  41. R. Askey, J. Wilson, Some basic hypergeometric polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc. 54(319), iv+55 (1985)

    Google Scholar 

  42. N.M. Atakishiyev, On a one-parameter family of q-exponential functions. J. Phys. A Math. Gen. 29(10), L223–L227 (1996)

    Google Scholar 

  43. M.K. Atakishiyeva, N.M. Atakishiyev, On continuous q-Hermite polynomials and the classical Fourier transform. J. Phys. A Math. Theor. 41(12), 9 (2008)

    Google Scholar 

  44. F. Atici, P. Eloe, Fractional q-calculus on time scale. J. Nonlinear Math. Phys. 14(3), 333–344 (2007)

    Google Scholar 

  45. F.V. Atkinson, Discrete and Continuous Boundary Problems, vol. 8 of Mathematics in Science and Engineering (Academic, New York, 1964)

    Google Scholar 

  46. B. Aulbach, S. Hilger, A Unified Approach to Continuous and Discrete Dynamics. Qualitative Theory of Differential Equations (Szegad, 1988), vol. 53 of Colloq. Math. Soc. János Bolayi (North-Holland, Amsterdam, 1990), pp. 37–56

    Google Scholar 

  47. E.W. Barnes, A new development of the theory of the hypergeometric functions. Proc. Lond. Math. Soc. s2–6(1), 141–177 (1908)

    Google Scholar 

  48. D.C. Barnett, R.G. Halburd, W. Morgan, R.J. Korhonen, Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations. Proc. R. Soc. Edinb. Sect. A 137(3), 457–474 (2007)

    Google Scholar 

  49. M.A. Bassam, Some properties of Holmgren-Riez transform. Ann. Scuola Norm. Sup. Pisa 15(3), 1–24 (1961)

    Google Scholar 

  50. M.B. Bekker, M.J. Bohner, A.N. Herega, H. Voulov, Spectral analysis of a q-difference operator. J. Phys. A Math. Theor. 43(14) (2010) pp. 15

    Google Scholar 

  51. M.S. Ben Hammouda, A. Nemri, Polynomial expansions for solutions of higher-order q-Bessel heat equation. Tamsui Oxf. J. Math. Sci. 24(2), 153–174 (2008)

    Google Scholar 

  52. Y.M. Berezansky, Z.G. Shaftl, G.F. US, Functional Analysis, vol. I (Birkhäuser, Basel, 1996)

    Google Scholar 

  53. W. Bergweiler, W.K. Hayman, Zeros of solutions of a functional equation. Comput. Meth. Funct. Theor. 3(1–2), 55–78 (2003)

    Google Scholar 

  54. N. Bettaibi, R.H. Bettaieb, q-analogue of the Dunkl transform on the real line. Tamsui Oxf. J. Math. Sci. 26(2), 178–206 (2009)

    Google Scholar 

  55. J.P. Bezivin, Sur les équations fonctionelles p-adiques aux q-différences. Collect. Math. 43(2), 125–140 (1992)

    Google Scholar 

  56. G.D. Birkhoff, Note on a canonical form for the linear q-difference system. Proc. Natl. Acad. Sci. 27, 218–222 (1941)

    Google Scholar 

  57. R.P. Boas, Entire Functions (Academic, New York, 1954)

    Google Scholar 

  58. L. Boelen, C. Smet, W. Van Assche, q-discrete painlevé equations for recurrence coefficients of modified q-Freud orthogonal polynomials. J. Difference Equ. Appl. 16(1), 37–53 (2010)

    Google Scholar 

  59. M. Bohner, G. Sh. Guseinov, The convolution on time scales. Abstr. Appl. Anal. (2007). Art. ID 58373 pp. 24

    Google Scholar 

  60. M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications (Birkhäuser, Boston, 2001)

    Google Scholar 

  61. M. Bohner, A. Peterson, Laplace transform and z-transform: Unification and extension methods. Appl. Anal. 9(1), 151–157 (2002)

    Google Scholar 

  62. F. Bouzeffour, Interpolation of entire functions, product formula for basic sine function. J. Nonlinear Math. Phys. 13(2), 293–301 (2006)

    Google Scholar 

  63. F. Bouzeffour, Inversion formulas for q-Riemann-Liouville and q-Weyl transforms. J. Math. Anal. Appl. 336(2), 833–848 (2007)

    Google Scholar 

  64. K. Brahim, R. Quanes, Some applications of the q-Mellin transform. Tamsui Oxf. J. Math. Sci. 26(3), 335–343 (2010)

    Google Scholar 

  65. T.J.I’A. Bromwich, An Introduction to the Theory of Infinite Series, 1st edn. (Macmillan, London, 1908)

    Google Scholar 

  66. J. Bustoz, J.L. Cardoso, Basic analog of Fourier series on a q-linear grid. J. Approx. Theor. 112(1), 134–157 (2001)

    Google Scholar 

  67. J. Bustoz, S.K. Suslov, Basic analog of Fourier series on a q-quadratic grid. Meth. Appl. Anal. 5(1), 1–38 (1998)

    Google Scholar 

  68. P.L. Butzer, U. Westphal, in An Introduction to Fractional Calculus, Chap. I. Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000), pp. 3–85

    Google Scholar 

  69. M. Caputo, Elasticitàe Dissipazione (Zanichelli, Bologna, 1969)

    Google Scholar 

  70. M. Caputo, Linear models of dissipation whose Q is almost frequency independent–II. Fract. Calc. Appl. Anal. 11(1), 4–14 (2008). Reprinted from Geophys. J. R. Astron. Soc. 13(5), 529–539 (1976)

    Google Scholar 

  71. R.D. Carmichael, The general theory of linear q-difference equations. Am. J. Math. 34, 147–168 (1912)

    Google Scholar 

  72. R.D. Carmichael, Linear difference equations and their analytic solutions. Trans. Am. Math. Soc. 12(2), 99–134 (1912)

    Google Scholar 

  73. R.D. Carmichael, On the theory of linear difference equations. Am. J. Math. 35(2), 163–182 (1913)

    Google Scholar 

  74. B.Q. Chen, X.Z. Chen, S. Li, Properties on solutions of some q-difference equations. Acta Math. Sin. (Engl. Ser.) 26(10), 1877–1886 (2010)

    Google Scholar 

  75. Y. Chen, M.E.H. Ismail, K.A. Muttalib, Asymptotics of basic Bessel functions and q-Laguerre polynomials. J. Comp. Appl. Math. 54, 263–273 (1994)

    Google Scholar 

  76. K. Chung, W. Chung, S. Nam, H. Kang, New q-derivative and q-logarithm. Int. J. Theoret. Phys. 33(10), 2019–2029 (1993)

    Google Scholar 

  77. J.A. Cochran, The Analysis of Linear Integral Equations (McGraw Hill, New York, 1972)

    Google Scholar 

  78. E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955)

    Google Scholar 

  79. T. Craven, G. Csordas, The Fox–Wright functions and Laguerre multiplier sequences. J. Math. Anal. Appl. 314(1), 109–125 (2006)

    Google Scholar 

  80. A. De Sole, V.G. Kac, On integral representations of q-gamma and q-beta functions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16(1), 11–29 (2005)

    Google Scholar 

  81. J.B. Diaz, T.J. Osler, Differences of fractional order. Math. Comp. 28(125), 185–202 (1974)

    Google Scholar 

  82. K. Diethelm, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes of Mathematics (Springer, Berlin, 2010)

    Google Scholar 

  83. K. Diethelm, N.J. Ford, A.D. Freed, Yu. Luchko, Algorithms for the fractional calculus: A selection of numerical methods. Comput. Meth. Appl. Mech. Eng. 194(6–8), 743–773 (2005)

    Google Scholar 

  84. M.M. Djrbashian, Integral Transforms and Representations of Functions in the Complex Domain (Nauka, Moscow, 1962). In Russian

    Google Scholar 

  85. M.M. Djrbashian, Harmonic Analysis and Boundary Value Problem in the Complex Domain (Birkhäuser, Basel, 1993)

    Google Scholar 

  86. A. Dobrogowska, A. Odzijewicz, Second order q-difference equations solvable by factorization method. J. Comput. Appl. Math. 193(1), 319–346 (2006)

    Google Scholar 

  87. A. Dobrogowska, A. Odzijewicz, Solutions of the q-deformed Schrödinger equation for special potentials. J. Phys. A Math. Theor. 40(9), 2023–2036 (2007)

    Google Scholar 

  88. M.S.P. Eastham, Theory of Ordinary Differential Equations (Van Nostrand, Reinhold, 1970)

    Google Scholar 

  89. A.M.I. El-Guindy, Z.S.I. Mansour, Functional definitions for q-analogues of Eulerian functions and applications. Aquationes Math. DOI:10.1007/s00010-012-0141-2

    Google Scholar 

  90. A.M.A. El-Sayed, Fractional order evolution equations. J. Fract. Calc. 7, 89–100 (1995)

    Google Scholar 

  91. A.M.A. El-Sayed, Multivalued fractional differential equations. Appl. Math. Comput. 86(1), 15–25 (1995)

    Google Scholar 

  92. M. El-Shahed, A. Salem, q-analogue of Wright function. Abstr. Appl. Anal. (2008). Art. ID 962849 pp. 11

    Google Scholar 

  93. A. Erdélyi, On some functional transformations. Rend. Sem. Mat. Univ. Politec. Torino 10, 217–234 (1951)

    Google Scholar 

  94. A. Erdélyi, H. Kober, Some remarks on Hankel transforms. Quart. J. Math. 11, 212–221 (1940). Oxford Series

    Google Scholar 

  95. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. III. (McGraw-Hill, New York, 1955)

    Google Scholar 

  96. A. Erdélyi, I.N. Sneddon, Fractional integration and dual integral equations. Can. J. Math. 14(4), 685–693 (1962)

    Google Scholar 

  97. V.V. Eremin, A.A. Meldianov, The q-deformed harmonic oscillator, coherent states, and the uncertainty relation. Theor. Math. Phys. 147(2), 709–715 (2006). Translation from Teor. Mat. Fiz. 147(2), 315–322 (2006)

    Google Scholar 

  98. H. Exton, q-Hypergeometric Functions and Applications (Ellis-Horwood, Chichester, 1983)

    Google Scholar 

  99. H. Exton, Basic Sturm-Liouville theory. Rev. Tecn.  Fac.  Ingr.  Univ.  Zulia 1, 85–100 (1992)

    Google Scholar 

  100. R.A.C. Ferreira, Nontrivial solutions for fractional q-difference boundary value problems. Electron. J. Qual. Theor. Differ. Equat. 10(70), 10 (2010)

    Google Scholar 

  101. C.M. Field, N. Joshi, F.W. Nijhoff, q-difference equations of KdV type and Chazy-type second-degree difference equations. J. Phys. A Math. Theor. 41(33), 13 (2008)

    Google Scholar 

  102. A. Fitouhi, N. Bettaibi, K. Brahim, The Mellin transform in quantum calculus. Constr. Approx. 23(3), 305–323 (2006)

    Google Scholar 

  103. A. Fitouhi, A. Safraoui, Paley-Wiener theorem for the q 2-Fourier-Rubin transform. Tamsui Oxf. J. Math. Sci. 26(3), 287–304 (2010)

    Google Scholar 

  104. R. Floreanini, J . LeTourneux, L. Vinet, More on the q-oscillator algebra and q-orthogonal polynomials. J. Phys. A Math. Gen. 28, L287–L293 (1995)

    Google Scholar 

  105. R. Floreanini, L. Vinet, A model for the continuous q-ultraspherical polynomials. J. Math. Phys. 36, 3800–3813 (1995)

    Google Scholar 

  106. R. Floreanini, L. Vinet, More on the q-oscillator algebra and q-orthogonal polynomials. J. Phys. A 28, L287–L293 (1995)

    Google Scholar 

  107. M. Foupouagnigni, W. Koepf, A. Ronveaux, Fourth-order difference equation for the associated classical orthogonal polynomials. J. Comput. Appl. Math. 92(2), 103–108 (1998)

    Google Scholar 

  108. M. Foupouagnigni, A. Ronveaux, W. Koepf, Fourth order q-difference equation for the first associated of the q-classical orthogonal polynomials. J. Comput. Appl. Math. 101(1–2), 231–236 (1999)

    Google Scholar 

  109. C. Fox, The asymptotic expansion of generalized hypergeometric functions. Proc. Lond. Math. Soc. S2–27(1), 389–400 (1928)

    Google Scholar 

  110. M.C. Gaer, L.A. Rubel, in The Fractional Derivative and Entire Functions. Fractional Calculus and Its Applications. Proceedings of International Conference, University of New Haven, West Haven, CT, 1974, vol. 457 of Lecture Notes in Math. (Springer, Berlin, 1975), pp. 171–206

    Google Scholar 

  111. Lj. Gajić, B. Stanković, Some properties of Wright’s function. Publ. Inst. Math. (Beograd) (N.S.) 20(34), 91–98 (1976)

    Google Scholar 

  112. L. Galué, Generalized Erdélyi-Kober fractional q-integral operator. Kuwait J. Sci. Eng. 36(2A), 21–34 (2009)

    Google Scholar 

  113. G. Gasper, M. Rahman, Basic Hypergeometric Series, 2nd edn. (Cambridge university Press, Cambridge, 2004)

    Google Scholar 

  114. R. Gorenflo, A.A. Kilbas, S. Rogozin, On the generalized Mittag-Leffler type functions. Integr. Transforms Spec. Funct. 7(3–4), 215–224 (1998)

    Google Scholar 

  115. R. Gorenflo, Y. Luchko, F. Mainardi, Analytical properties and applications of the Wright function. Fract. Calc. Appl. Anal. 2(4), 383–414 (1999)

    Google Scholar 

  116. R. Gorenflo, Yu. Luchko, J. Loutchko, Computation of the Mittag-Leffler function E α, β(z) and its derivative. Frac. Calc. Appl. Anal. 5(4), 491–518 (2002)

    Google Scholar 

  117. R. Gorenflo, F. Mainardi, in Fractional Calculus: Integral and Differential Equations of Fractional Order, ed. by A. Carpinteri, F. Mainardi. Fractals and Fractional Calculus in Continuum Mechanics, vol. 378 of CISM Courses and Lecture Notes (Springer, New York, 1997), pp. 223–276

    Google Scholar 

  118. R. Gorenflo, S. Vessella, in Abel Integral Equations: Analysis and Applications. Lecture Notes in Mathematics, vol. 1461 (Springer, Berlin, 1991)

    Google Scholar 

  119. R.W. Gray, C.A. Nelson, A completeness relation for the q-analogue coherent states by q-integration. J. Phys. A. 23, L945–L950 (1990)

    Google Scholar 

  120. A.K. Grünwald, Über begrenzte derivationen und deren anwedung. Z. Math. Phys. 12, 441–480 (1867)

    Google Scholar 

  121. I. Győri, G. Ladas, Oscillation Theory of Delay Differential Equations with Applications (Clarendon press, Oxford, 1991)

    Google Scholar 

  122. W. Hahn, Beiträge zur Theorie der Heineschen Reihen. Die 24 Integrale der Hypergeometrischen q-Differenzengleichung. Das q-Analogon der Laplace-transformation (German). Math. Nachr. 2, 340–379 (1949)

    Google Scholar 

  123. W. Hahn, Beiträge zur Theorie der Heineschen Reihen (German). Math. Nachr. 2, 340–379 (1949)

    Google Scholar 

  124. W. Hahn, Über die zerlegung einer klasse von polynomen in irreduzible faktoren (German). Math. Nachr. 3, 257–294 (1950)

    Google Scholar 

  125. W. Hahn, Über uneigentliche Lösungen linearer geometrischer Differenzengleichungen (German). Math. Ann. 125, 67–81 (1952)

    Google Scholar 

  126. J.W. Hanneken, B.N.N. Achar, R. Puzio, D.M. Vaught, Properties of the Mittag-Leffler function for negative alpha. Phys. Scr. T136, 5 (2009)

    Google Scholar 

  127. H.J. Haubold, A.M. Mathai, R.K. Saxena, Mittag-Leffler functions and their applications. J. Appl. Math. 2011, 51 (2011)

    Google Scholar 

  128. W.K. Hayman, Subharmonic Functions, vol. II (Academic, London, 1989)

    Google Scholar 

  129. W.K. Hayman, in On the Zeros of a q-Bessel Function, ed. by M. Agranovsky, L. Karp, D. Shoikhet. Complex Analysis and Dynamical Systems II: A Conference in Honor of Professor Lawrence Zalcman’s Sixtieth Birthday, June 9–12, 2003, vol. 382 of Contemp. Math. (Amer. Math. Soc., Providence, 2005), pp. 205–216

    Google Scholar 

  130. E. Heine, Über die reihe... J. Reine Angew. Math. 32, 210–212 (1846)

    Google Scholar 

  131. E. Heine, Untersuchngen über die reihe... J. Reine Angew. Math. 34, 285–328 (1847)

    Google Scholar 

  132. E. Heine, Handbuch der Kugelfunctionen, Theorie und Anwendungen, vol. 1 (G. Reimer, Berlin, 1878)

    Google Scholar 

  133. E. Heine, Handbuch der Kugelfunctionen, Theorie und Anwendungen, vol. 2 (G. Reimer, Berlin, 1881)

    Google Scholar 

  134. N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 45(5), 765–771 (2006)

    Google Scholar 

  135. S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math. 18(1–2), 18–56 (1990)

    Google Scholar 

  136. C.L. Ho, On the use of Mellin transform to a class of q-difference-differential equations. Phys. Lett. A 268(4–6), 217–223 (2000)

    Google Scholar 

  137. P. Humbert, R.P. Agarwal, Sur la fonction de Mittag–Leffler et quelques-unes de ses généralisations (French). Bull. Sci. Math. 77(2), 180–185 (1953)

    Google Scholar 

  138. A. Huseynov, E. Bairamov, An eigenvalue problem for quadratic pencils of q-difference equations and its applications. Appl. Math. Lett. 22(4), 521–527 (2009)

    Google Scholar 

  139. J.I. Hutchinson, On a remarkable class of entire functions. Trans. Am. Math. Soc. 25(3), 325–332 (1923)

    Google Scholar 

  140. M.E.H. Ismail, The basic Bessel functions and polynomials. SIAM J. Math. Anal. 12(3), 454–468 (1981)

    Google Scholar 

  141. M.E.H. Ismail, The zeros of basic Bessel functions, the functions J ν + α(x) and associated orthogonal polynomials. J. Math. Anal. Appl. 86(1), 11–19 (1982)

    Google Scholar 

  142. M.E.H. Ismail, On Jackson’s third q-Bessel function. preprint (1996)

    Google Scholar 

  143. M.E.H Ismail, Classical and Quantum Orthogonal Polynomials in One Variable (Cambridge University Press, Cambridge, 2005)

    Google Scholar 

  144. M.E.H. Ismail, S.J. Johnston, Z.S. Mansour, Structure relations for q-polynomials and some applications. Appl. Anal. 90(3–4), 747–767 (2011)

    Google Scholar 

  145. M.E.H. Ismail, L. Lorch, M.E. Muldoon, Completely monotonic functions associated with the gamma functions and its q-analogues. J. Math. Anal. Appl. 116, 1–9 (1986)

    Google Scholar 

  146. M.E.H. Ismail, Z.S.I. Mansour, q-analogues of Freud weights and nonlinear difference equations. Adv. Appl. Math. 45(4), 518–547 (2010)

    Google Scholar 

  147. M.E.H. Ismail, M. Rahman, Inverse operators, q-fractional integrals and q-Bernoulli polynomials. J. Approx. Theor. 114(2), 269–307 (2002)

    Google Scholar 

  148. M.E.H. Ismail, D. Stanton, Applications of q-Taylor theorems. J. Comp. Appl. Math. 153(1–2), 259–272 (2003)

    Google Scholar 

  149. M.E.H. Ismail, D. Stanton, q-Taylor theorems, polynomial expansions, and interpolation of entire functions. J. Approx. Theor. 123(1), 125–146 (2003)

    Google Scholar 

  150. M.E.H. Ismail, A.I. Zayed, A q-analogue of the Whittaker-Shannon-Koteĺnikov sampling theorem. Proc. Am. Math. Soc. 131(2), 3711–3719 (2003)

    Google Scholar 

  151. M.E.H. Ismail, C. Zhang, Zeros of entire functions and a problem of Ramanujan. Adv. Math. 209(1), 363–380 (2007)

    Google Scholar 

  152. M.E.H. Ismail, R. Zhang, Diagonalization of certain integral operators. Adv. Math. 109(1), 1–33 (1994)

    Google Scholar 

  153. F.H. Jackson, On generalized functions of Legendre and Bessel. Trans. Roy. Soc. Edinb. 41, 1–28 (1903)

    Google Scholar 

  154. F.H. Jackson, A basic-sine and cosine with symbolical solutions of certain differential equations. Proc. Edinb. Math. Soc. 22, 28–38 (1903/1904)

    Google Scholar 

  155. F.H. Jackson, A generalization of the function Γ(n) and x n. Proc. Roy. Soc. Lond. 74, 64–72 (1904)

    Google Scholar 

  156. F.H. Jackson, The applications of basic numbers to Bessel’s and Legendre’s equations (second paper). Proc. Lond. Math. Soc. 3(2), 1–23 (1905)

    Google Scholar 

  157. F.H. Jackson, The basic gamma function and elliptic functions. Proc. Roy. Soc. A 76, 127–144 (1905)

    Google Scholar 

  158. F.H. Jackson, On q-functions and a certain difference operator. Trans. Roy. Soc. Edinb. 46, 64–72 (1908)

    Google Scholar 

  159. F.H. Jackson, A q-form of Taylor’s theorem. Messenger Math. 38, 62–64 (1909)

    Google Scholar 

  160. F.H. Jackson, On q-definite integrals. Quart. J. Pure Appl. Math. 41, 193–203 (1910)

    Google Scholar 

  161. F.H. Jackson, The q-integral analogous to Borel’s integral. Messenger Math. 47, 57–64 (1917)

    Google Scholar 

  162. A. Jirari, Second-order Sturm Liouville difference equations and orthogonal polynomials. Mem. Am. Math. Soc. 115(542)x, pp. 138 (1995)

    Google Scholar 

  163. V. Kac, P. Cheung, Quantum Calculus (Springer, New York, 2002)

    Google Scholar 

  164. H.H. Kairies, M.E. Muldoon, Some characterizations of q-factorial functions. Aequationes Mathematicae 25, 67–76 (1982)

    Google Scholar 

  165. H. Karabulut, Distributed Gaussian polynomials as q-oscillator eigenfunctions. J. Math. Phys. 47(1), 13 (2006)

    Google Scholar 

  166. O.M. Katkova, A.M. Vishnyakova, A sufficient condition for a polynomial to be stable. J. Math. Anal. Appl. 347(1), 81–89 (2008)

    Google Scholar 

  167. A.A. Kilbas, Fractional calculus of the generalized Wright function. Fract. Calc. Appl. Anal. 8(2), 113–126 (2005)

    Google Scholar 

  168. A.A. Kilbas, M. Saigo, R.K Saxena, Generalized Mittag-Leffler functions and generalized fractional calculus operator. Integr. Transforms Spec. Funct. 15(1), 31–49 (2004)

    Google Scholar 

  169. A.A. Kilbas, H.M. Srivastava, J.I. Trujillo, Theory and Applications of Fractional Differential Equations, 1st edn. (Elsevier, London, 2006)

    Google Scholar 

  170. Y.O. Kim, A proof of the Pólya-Wiman conjecture. Proc. Am. Math. Soc. 109(4), 1045–1052 (1990)

    Google Scholar 

  171. W.A. Kirk, M.A. Khamsi, An Introduction to Metric Spaces and Fixed Point Theory (Wiley, New York, 2001)

    Google Scholar 

  172. V.S. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. Higher transcendental functions and their applications. J. Comput. Appl. Math. 118(1–2), 241–259 (2000)

    Google Scholar 

  173. H. Kober, On fractional integrals and derivatives. Quart. J. Math. Oxford Ser. 11, 193–211 (1940)

    Google Scholar 

  174. H.T. Koelink, R.F. Swarttouw, On the zeros of the Hahn-Exton q-Bessel function and associated q-Lommel polynomials. J. Math. Anal. Appl. 186(3), 690–710 (1994)

    Google Scholar 

  175. A.N. Kolmogorov, S.V. Fomin, Introductory Real Analysis (Dover Publications, New York, 1957)

    Google Scholar 

  176. T.H. Koornwinder, R.F. Swarttouw, On q-analogues of Fourier and Hankel transforms. Trans. Am. Math. Soc. 333(1), 445–461 (1992)

    Google Scholar 

  177. V.P. Kostov, B. Shapiro, Hardy-Petrovitch-Hutchinson’s problem and partial theta function. arXiv:1106.6262v1[math. CA] (2011)

    Google Scholar 

  178. J.L. Lavoie, T.J. Osler, R. Tremblay, Fractional derivative and special functions. SIAM Rev. 18(2), 240–268 (1976)

    Google Scholar 

  179. C.W. Leibniz, Mathematische Schriften (Georg Olms Verlagsbuchhandlung, Hildesheim, 1962)

    Google Scholar 

  180. A.V. Letinkov, Theory of differentiation of fractional order. Mat. Sb. 3, 1–68 (1868)

    Google Scholar 

  181. B.Ja. Levin, Distribution of Zeros of Entire Functions, vol. 5 of Translations of Mathematical Monographs (Amer. Math. Soc., Providence, 1980)

    Google Scholar 

  182. B.M. Levitan, I.S. Sargsjan, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators (Amer. Math. Soc., Providence, 1975)

    Google Scholar 

  183. B.M. Levitan, I.S. Sargsjan, Sturm Liouville and Dirac Operators (Kluwer Academic, Dordrecht, 1991)

    Google Scholar 

  184. X. Li, C. Zhang, Existence of analytic solutions to analytic nonlinear q-difference equations. J. Math. Anal. Appl. 375(2), 412–417 (2011)

    Google Scholar 

  185. M.J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge University Press, New York, 1960)

    Google Scholar 

  186. J. Liouville, Mèmoire sur le calcul des différentielles à indices quelconques. J. de I’Ecole Polytechnique 13, 71–162 (1832)

    Google Scholar 

  187. J.E. Littelwood, On the asymptotic approximation to integral functions of order zero. Proc. Lond. Math. Soc. 5(2), 361–410 (1907)

    Google Scholar 

  188. H.K. Liu, Application of a differential transformation method to strongly nonlinear damped q-difference equations. Comput. Math. Appl. 61(9), 2555–2561 (2011)

    Google Scholar 

  189. Y.K. Liu, The linear q-difference equation y(x) = ay(qx) + f(x). Appl. Math. Lett. 8(1), 15–18 (1995)

    Google Scholar 

  190. Z.G. Liu, Two q-difference equations and q-operator identities. J. Differ. Equat. Appl. 16(11), 1293–1307 (2010)

    Google Scholar 

  191. B. López, J.M. Marco, J. Parcet, Taylor series for the Askey-Wilson operator and classical summation formulas. Proc. Am. Math. Soc. 134(8), 2259–2270 (2006)

    Google Scholar 

  192. D.Q. Lu, q-difference equation and the Cauchy operator identities. J. Math. Anal. Appl. 359(1), 265–274 (2009)

    Google Scholar 

  193. Yu. Luchko, Asymptotics of zeros of the Wright function. Z. Anal. Anwendungen 19(2), 583–595 (2000)

    Google Scholar 

  194. I. Lutzenko, V. Spiridonov, A. Zhedanov, On the spectrum of a q-scillator with a linear interaction. Phys. Lett. A 204(3–4), 236–242 (1995)

    Google Scholar 

  195. H. Mabrouk, q-heat operator and q-Poisson’s operator. Fract. Calc. Appl. Anal. 9(3), 265–286 (2006)

    Google Scholar 

  196. J.T. Machado, V. Kiryakova, F. Mainardi, A poster about the recent history of fractional calculus. Fract. Calc. Appl. Anal. 13(3), 329–334 (2010)

    Google Scholar 

  197. J.T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus. Comm. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)

    Google Scholar 

  198. N.I. Mahmudov, P. Sabancıgil, q-parametric Bleimann Butzer and Hahn operators. J. Inequal. Appl. 2008, 15 (2008) (Art. ID 816367)

    Google Scholar 

  199. F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9(6), 23–28 (1996)

    Google Scholar 

  200. F. Mainardi, R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118(1–2), 283–299 (2000)

    Google Scholar 

  201. F. Mainardi, G. Pagnini, The Wright functions as solutions of the time-fractional diffusion equation. Appl. Math. Comput. 141(1), 51–62 (2003)

    Google Scholar 

  202. H.L. Manocha, B.L. Sharma, Fractional derivatives and summation. J. Indian Math. Soc. 38(1–4), 371–382 (1974)

    Google Scholar 

  203. Z.S.I. Mansour, q-Difference Equations. Master’s thesis, Faculty of Science, Cairo University, Giza, Egypt, 2001

    Google Scholar 

  204. Z.S.I. Mansour, On q-Difference and Fractional Equations and Their Solutions. PhD thesis, Faculty of Science, Cairo University, Giza, Egypt, 2006

    Google Scholar 

  205. Z.S.I. Mansour, Linear sequential q-difference equations of fractional order. Fract. Calc. Appl. Anal. 12(2), 160–178 (2009)

    Google Scholar 

  206. Z.S.I. Mansour, Generalizations of fractional q-Leibniz formulae. submitted (2012)

    Google Scholar 

  207. Z.S.I. Mansour, Generalizations of Rubin’s q 2-fourier transform and q-difference operator. submitted (2012)

    Google Scholar 

  208. V.A. Marchenko, Sturm–Liouville Operators and Applications (Birkhäuser, Basel, 1986)

    Google Scholar 

  209. T.E. Mason, On properties of the solutions of linear q-difference equations with entire function coefficients. Am. J. Math. 37(4), 439–444 (1915)

    Google Scholar 

  210. A. Matsuo, Jackson integrals of Jordan-Pochhammer type and quantum Knizhnik-Zamolodchikov equations. Comm. Math. Phys. 151(2), 263–273 (1993)

    Google Scholar 

  211. A.C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions (Pitman Research Notes in Mathematics, London, 1979)

    Google Scholar 

  212. H. Mellin, Om definita integraler. Acta Societatis Scientiarum Fennicae 20(7), 1–39 (1895)

    Google Scholar 

  213. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)

    Google Scholar 

  214. R. Mishra, On certain generalized q-Laplace and Stieltjes transforms. J. Math. Sci. 5(2), 22–37 (1970)

    Google Scholar 

  215. R. Mishra, A few inversion formulae for generalized q-Laplace transforms. Indian J. Math. 14(3), 157–172 (1972)

    Google Scholar 

  216. R. Mishra, Applications of fractional q-integration on the theory of a generalized q-Stieltjes transform. Bull. Calcutta. Math. Soc. 69(5), 247–254 (1977)

    Google Scholar 

  217. R. Mishra, Application of generalized q-Laplace transforms to the solutions of certain q-integral equations. Indian J. Math. 23(1–3), 115–120 (1981)

    Google Scholar 

  218. R. Mishra, On a relation between a q-Laplace transform and two generalized q-Laplace transforms. Proc. Natl. Acad. Sci. India Sect. A 53(4), 379–385 (1983)

    Google Scholar 

  219. G. Mittag-Leffler, Sur l’intégrale de Laplace-Abel. C. R. Acad. Sci. Paris (Ser. II) 136, 937–939 (1902)

    Google Scholar 

  220. G. Mittag-Leffler, Sur la nouvelle fonction E α(x). C. R. Acad. Sci. Paris 137, 554–558 (1903)

    Google Scholar 

  221. G. Mittag-Leffler, Une généralisation de l’intégrale de Laplace-Abel. C. R. Acad. Sci. Paris (Ser. II) 137, 537–539 (1903)

    Google Scholar 

  222. G. Mittag-Leffler, Sopra la funzione E α(x). R. Accad. Lincei, Rend. (Ser. V) 13, 3–5 (1904)

    Google Scholar 

  223. G. Mittag-Leffler, Sur la reprwesentation analytique dune branche uniforme dune fonction monogéne. Acta Math. 29, 101–182 (1905)

    Google Scholar 

  224. D.S. Moak, The q-gamma function for q > 1. Aequationes Mathematicae 20(2–3), 278–285 (1980)

    Google Scholar 

  225. A. Nemri, A. Fitouhi, Polynomial expansions for solution of wave equation in quantum calculus. Matematiche 65(1), 73–82 (2010)

    Google Scholar 

  226. Y. Nikolova, L. Boyadjiev, Integral transform method to solve a time-space fractional diffusion equation. Fract. Calc. Appl. Anal. 13(1), 57–67 (2010)

    Google Scholar 

  227. K.B. Oldham, J. Spanier, The Fractional Calculus (Academic, London, 1974)

    Google Scholar 

  228. T.J. Osler, Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM J. Appl. Math. 18(3), 658–674 (1970)

    Google Scholar 

  229. T.J. Osler, Fractional derivatives and Leibniz rule. Am. Math. Mon. 78(6), 645–649 (1971)

    Google Scholar 

  230. T.J. Osler, A further extension of the Leibniz rule to fractional derivatives and its relation to Parserval’s formula. SIAM J. Math. Anal. 3(1), 1–16 (1972)

    Google Scholar 

  231. T.J. Osler, The integral analogue of the Leibniz rule. Math. Comp. 26(120), 903–915 (1972)

    Google Scholar 

  232. R.B. Paris, D. Kaminski, Asymptotics and Mellin-Barnes Integrals (Cambridge University Press, Cambridge, 2001)

    Google Scholar 

  233. I. Podlubny, in Solution of Linear Fractional Differential Equations with Constant Coefficients, ed. by P. Rusev, I. Dimovski, V. Kiryakova. Transform Methods and Special Functions (SCT Publications, Singapore, 1995), pp. 217–228

    Google Scholar 

  234. I. Podlubny, Fractional Differential Equations (Academic, New York, 1999)

    Google Scholar 

  235. I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5(4), 367–386 (2002)

    Google Scholar 

  236. I. Podlubny, A.M.A. El-Sayed, On two definitions of fractional derivatives. UEF-03-96, Inst. Exp. Phys, Slovak Acad. Sci., 1996. ISBN 80-7099-252-2

    Google Scholar 

  237. G. Pólya, Über die Nullstellengewisser ganzer Functions(German). Math. Z. 2(3–4), 352–383 (1918)

    Google Scholar 

  238. G. Pólya, G. Szegö, Problems and Theorems in Analysis, vol. I (Springer, New York, 1972). Translated from the German by D. Aeppli Die Grundlehren der mathematischen Wissenschaften, Band 193

    Google Scholar 

  239. A.Yu. Popov, On the spectral values of a boundary value problem and the zeros of Mittag-Leffler functions. Differ. Equat. 38(5), 642–653 (2002). Translation from Differ. Uravn. 38(5), 611–621 (2002)

    Google Scholar 

  240. A.Yu. Popov, On zeroes of Mittag–Leffler functions with parameter ρ < 1 ∕ 2. Anal. Math. 32(3), 207–246 (2006)

    Google Scholar 

  241. A.Yu. Popov, A.M. Sedletskiĭ, Distribution of zeros of the Mittag-Leffler function. Dokl. Math. 67(3), 336–339 (2003). Translation from Dokl. Akad. Nauk 390(2), 165–168 (2003)

    Google Scholar 

  242. E.L. Post, Generalized differentiation. Trans. Am. Math. Soc. 32, 723–781 (1930)

    Google Scholar 

  243. A.D. Poularikas (ed.), The Transforms and Applications Handbook, 2nd edn. The Electrical Engineering Handbook Series (CRC Press, FL, 2000)

    Google Scholar 

  244. T.R. Prabhakar, A singular integral equation with a generalized Mittag–Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)

    Google Scholar 

  245. S.D. Prasad, Certain q-transforms. Math. Scand. 30, 324–330 (1972)

    Google Scholar 

  246. A.V. Pskhu, On the real zeros of functions of Mittag-Leffler type. Math. Notes 77(3–4), 546–552 (2005). Translation from Mat. Zametki 77(4), 592–599 (2005)

    Google Scholar 

  247. S.D. Purohit, Summation formulae for basic hypergeometric functions via fractional q-calculus. Mathematica (Catania) 64(1), 67–75 (2009)

    Google Scholar 

  248. S.D. Purohit, On a q-extension of the Leibniz rule via Weyl type of q-derivative operator. Kyungpook Math. J. 50(4), 473–482 (2010)

    Google Scholar 

  249. S.D. Purohit, R.K. Raina, Generalized q-Taylor’s series and applications. Gen. Math. 18(3), 19–28 (2010)

    Google Scholar 

  250. S.D. Purohit, R.K. Yadav, On generalized fractional q-integral operators involving the q-Gauss hypergeometric function. Bull. Math. Anal. Appl. 2(4), 35–44 (2010)

    Google Scholar 

  251. M. Rahman, A note on the orthogonality of Jackson’s q-Bessel functions. Can. Math. Bull. 32(3), 369–376 (1989)

    Google Scholar 

  252. M. Rahman, Q.M. Tariq, Addition formulas for q-Legendre-type functions. Meth. Appl. Anal. 6(1), 3–20 (1999)

    Google Scholar 

  253. P.M. Rajković, M. Stanković, S.D. Marinković, Mean value theorems in q-calculus. Mat. Vesnik 54, 171–178 (2002)

    Google Scholar 

  254. P.M. Rajković, S.D. Marinković, M.S. Stanković, Fractional integrals and derivatives in q-calculus. Appl. Anal. Discrete Math. 1(1), 311–323 (2007)

    Google Scholar 

  255. P.M. Rajković, S.D. Marinković, M.S. Stanković, On q-analogues of Caputo derivative and Mittag–Leffler function. Fract. Calc. Appl. Anal. 10(4), 359–373 (2007)

    Google Scholar 

  256. J.P. Ramis, About the growth of entire functions solutions of linear algebraic q-difference equations. Ann. Fac. Sci. Toulouse Math. 1(6), 53–94 (1992)

    Google Scholar 

  257. C. Richard, On q-functional equations and excursion moments. Discrete Math. 309(1), 207–230 (2009)

    Google Scholar 

  258. E.Yu. Romanenko, Differential-difference equations reducible to difference and q-difference equations. Comput. Math. Appl. 42(3–5), 615–626 (2001)

    Google Scholar 

  259. P.G. Rooney, On the ranges of certain fractional integrals. Can. J. Math. 24, 1198–1216 (1972)

    Google Scholar 

  260. P.G. Rooney, A survey of mellin multipliers. fractional calculus. Res. Notes Math. 138, 176–187 (1985)

    Google Scholar 

  261. B. Ross (ed.), Fractional Calculus and Its Applications, vol. 457 of Lecture Notes of Mathematics. Proceeding of the International Conference held at the University of New Haven, June 15-16, 1974 (Springer, Berlin, 1975)

    Google Scholar 

  262. B. Ross, Fractional Calculus. An historical apologia for the development of a calculus using differentiation and antidifferentiation of non-integral orders. Math. Mag. 50(3), 115–122 (1977)

    Google Scholar 

  263. B. Ross, Origins of fractional calculus and some applications. Int. J. Math. Statist. Sci. 1(1), 21–34 (1992)

    Google Scholar 

  264. B. Ross, B.K. Sachdeva, The solution of certain integral equations by means of operators of arbitrary order. Am. Math. Mon. 97(6), 498–503 (1990)

    Google Scholar 

  265. R.L. Rubin, A q 2-analogue operator for q 2-analogue Fourier analysis. J. Math. Anal. Appl. 212(2), 571–582 (1997)

    Google Scholar 

  266. R.L. Rubin, Duhamel solutions of non-homogeneous q 2-analogue wave equations. Proc. Am. Math. Soc. 135(3), 777–785 (2007)

    Google Scholar 

  267. J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering (Springer, London, 2007)

    Google Scholar 

  268. E.B. Saff, A.D. Snider, Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, 3rd edn. (Prentice-Hall, Englewood Cliffs, 2003)

    Google Scholar 

  269. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications (Gordan and Breach Science Publisher, New York, 1993)

    Google Scholar 

  270. R.K. Saxena, G.C. Modi, S.L Kalla, A basic analogue of Fox’s H-function. Rev. Técn. Fac. Ingr. Univ. Zulia 6, 139–143 (1983). Special Issue

    Google Scholar 

  271. R.K. Saxena, R.K. Yadav, S.D. Purohit, Kober fractional q-integral operator of the basic analogue of the H-function. Rev. Téc. Ing. Univ. Zulia, ago 28(2), 154–158 (2005)

    Google Scholar 

  272. A.M. Sedletskiĭ, Asymptotic formulae for zeros of a function of Mittag-Leffler’s type. Anal. Math. 20(2), 117–132 (1994)

    Google Scholar 

  273. A.M. Sedletskiĭ, On the zeros of functions of Mittag–Leffler type. Math. Notes 68(5), 602–613 (2000). Translation from Mat. Zametki 68(5), 710–724 (2000)

    Google Scholar 

  274. S. Sergeev, Quantum curve in q-oscillator model. Int. J. Math. Math. Sci. 2006, 31 (2006) (Art. ID 92064)

    Google Scholar 

  275. G.E. Shilov, Linear Algebra (Dover, New York, 1971)

    Google Scholar 

  276. I.N. Sneddon, Mixed Boundary ValueProblems (North Holland, Amsterdam, 1966)

    Google Scholar 

  277. S.K. Suslov, Another addition theorem for the q-exponential function. J. Phys. A Gen. 33, L375–L380 (2000)

    Google Scholar 

  278. S.K. Suslov, Some expansions in basic Fourier series and related topics. J. Approx. Theor. 115, 289–353 (2002)

    Google Scholar 

  279. S.K. Suslov, An Introduction to Basic Fourier Series, vol. 9 of Kluwer Series Developments in Mathematics (Kluwer Academic Publishers, Boston, 2003)

    Google Scholar 

  280. R.F. Swarttouw, The Hahn-Exton q-Bessel Function. PhD thesis, The Technical University of Delft, 1992

    Google Scholar 

  281. R.F. Swarttouw, H.G. Meijer, A q-analogue of the Wronskian and a second solution of the Hahn-Exton q-Bessel difference equation. Proc. Am. Math. Soc. 120(3), 855–864 (1994)

    Google Scholar 

  282. J. Thomae, Beitrage zur theorie der durch die heinesche reihe …. J. Reine Angew. Math. 70, 258–281 (1869)

    Google Scholar 

  283. I.V. Tikhonov, Yu.S. Éĭdeĺman, An inverse problem for a differential equation in a Banach space and the distribution of zeros of an entire Mittag-Leffler function. Differ. Equat. 38(5), 669–677 (2002). Translation from Differ. Uravn. 38(5), 637–644 (2002)

    Google Scholar 

  284. E. Titchmarsh, Eigenfunction Expansions Associated with Second Order Differential Equations (Clarendon Press, Oxford, 1962)

    Google Scholar 

  285. W.J. Trjitzinsky, Analytic theory of linear q-difference equations. Acta Math. 61(1), 1–38 (1933)

    Google Scholar 

  286. T. Tsuda, On an integrable system of q-difference equations satisfied by the universal characters: Its lax formalism and an application to q-painlevé equations. Comm. Math. Phys. 293(2), 347–359 (2010)

    Google Scholar 

  287. L. Di Vizio, Local analytic classification of q-difference equations with | q |  = 1. J. Noncommut. Geom. 3(1), 125–149 (2009)

    Google Scholar 

  288. Y. Watanabe, Notes on the generalized derivative of Riemann–Liouville and its application to Leibniz’s formula. I and II. Tôhoku Math. J. 34, 8–41 (1931)

    Google Scholar 

  289. M. Welter, Interpolation of entire functions on regular sparse sets and q-Taylor series. J. Théor. Nombres Bordeaux 17(1), 397–404 (2005)

    Google Scholar 

  290. E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, 4th edn. (Cambridge University Press, London, 1927). Reprinted (1973)

    Google Scholar 

  291. D.V. Widder, Laplace Transform (Princeton University Press, Princeton, 1941)

    Google Scholar 

  292. A. Wiman, Über den fundamentalsatz in der teorie der funktionenE α(x). Acta Math. 29, 191–201 (1905)

    Google Scholar 

  293. E.M. Wright, On the coefficients of power series having exponential singularities. J. Lond. Math. Soc. 8, 71–79 (1933)

    Google Scholar 

  294. E.M. Wright, The asymptotic expansion of the generalized hypergeometric function. J. Lond. Math. Soc. s1–10(4), 286–293 (1935)

    Google Scholar 

  295. E.M. Wright, The asymptotic expansion of integral functions defined by Taylor series. Philos. Trans. Roy. Soc. Lond. Ser. A 238, 423–451 (1940)

    Google Scholar 

  296. E.M. Wright, The asymptotic expansion of the generalized hypergeometric function II. Proc. Lond. Math. Soc. 46(2), 389–408 (1940)

    Google Scholar 

  297. W. Wyss, The fractional diffusion equation. J. Math. Phys. 27(11), 2782–2785 (1986)

    Google Scholar 

  298. B. Xu, W. Zhang, Small divisor problem for an analytic q-difference equation. J. Math. Anal. Appl. 342(1), 694–703 (2008)

    Google Scholar 

  299. X. Zheng, Z. Chen, Some properties of meromorphic solutions of q-difference equations. J. Math. Anal. Appl. 361(2), 472–480 (2010)

    Google Scholar 

  300. A. Zygmund, Trigonometric Series, vol. II (Cambridge University Press, Cambridge, 1959)

    Google Scholar 

  301. M. Ismail, SIAM Rev. 16, 279–281 (1985)

    Google Scholar 

  302. W. Hahn, Math. Rev. 84i, 39002 (1982)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Science Department of Mathematics, Cairo University, Giza, Egypt

    Mahmoud H. Annaby

  2. Faculty of Science Department of Mathematics, King Saud University, Riyadh, Kingdom of Saudi Arabia

    Zeinab S. Mansour

Authors
  1. Mahmoud H. Annaby
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zeinab S. Mansour
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Annaby, M.H., Mansour, Z.S. (2012). q-Sturm–Liouville Problems. In: q -Fractional Calculus and Equations. Lecture Notes in Mathematics, vol 2056. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30898-7_3

Download citation

Publish with us

Policies and ethics

Search

Navigation