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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 q 2 - Fourier transforms are studied and their applications are investigated.

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1 Preliminaries.- 2 q-Difference Equations.- 3 q-Sturm Liouville Problems.- 4 Riemann-Liouville q-Fractional Calculi.- 5 Other q-Fractional Calculi.- 6 Fractional q-Leibniz Rule and Applications.- 7 q-Mittag-Leffler Functions.- 8 Fractional q-Difference Equations.- 9 Applications of q-Integral Transforms.

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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)

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