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 q2-*Fourier transforms are studied and their applications are investigated.