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.
