q-Fractional Calculus and Equations

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.