© 2012

q -Fractional Calculus and Equations


  • First detailed rigorous study of q-calculi

  • First detailed rigorous study of q-difference equations

  • First detailed rigorous study of q-fractional calculi and equations

  • Proofs of many classical unproved results are given

  • Illustrative examples and figures helps readers to digest the new approaches


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

Table of contents

  1. Front Matter
    Pages i-xix
  2. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 1-39
  3. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 41-71
  4. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 73-105
  5. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 107-146
  6. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 147-173
  7. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 175-199
  8. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 201-222
  9. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 223-270
  10. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 271-293
  11. Back Matter
    Pages 295-318

About this book


This nine-chapter monograph introduces a rigorous investigation of q-difference operators in standard and fractional settings. It starts with elementary calculus of q-differences and integration of Jackson’s type before turning to q-difference equations. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Regular  q-Sturm–Liouville theory is also introduced; Green’s function is constructed and the eigenfunction expansion theorem is given. The monograph also discusses some integral equations of Volterra and Abel type, as introductory material for the study of fractional q-calculi. Hence fractional q-calculi of the types Riemann–Liouville; Grünwald–Letnikov;  Caputo;  Erdélyi–Kober and Weyl are defined analytically. Fractional q-Leibniz rules with applications  in q-series are  also obtained with rigorous proofs of the formal  results of  Al-Salam-Verma, which remained unproved for decades. In working towards the investigation of q-fractional difference equations; families of q-Mittag-Leffler functions are defined and their properties are investigated, especially the q-Mellin–Barnes integral  and Hankel contour integral representation of  the q-Mittag-Leffler functions under consideration,  the distribution, asymptotic and reality of their zeros, establishing q-counterparts of Wiman’s results. Fractional q-difference equations are studied; existence and uniqueness theorems are given and classes of Cauchy-type problems are completely solved in terms of families of q-Mittag-Leffler functions. Among many q-analogs of classical results and concepts, q-Laplace, q-Mellin and q2-Fourier transforms are studied and their applications are investigated.


33D15, 26A33, 30C15, 39A13, 39A70 Basic Hypergeometric functions One variable calculus Zeros of analytics functions q$-difference equations

Authors and affiliations

  1. 1.Faculty of Science, Department of MathematicsCairo UniversityGizaEgypt
  2. 2.Faculty of Science, Department of MathematicsKing Saud UniversityRiyadhSaudi Arabia

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From the reviews:

“This monograph briefly introduces q-calculus … . The book is carefully and well written. Each chapter is introduced by an informative abstract. The bibliography is extensive and useful, and useful tables of formulas appear in appendices. This monograph is of interest to people who want to learn to do research in q-fractional calculus as well as to people currently doing research in q-fractional calculus.” (P. W. Eloe, Mathematical Reviews, April, 2013)