Kilbas A. H-Transforms. Theory and Applications 2004

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This book deals with integral transforms involving the so-called H-functions as kernels, or H-transforms, and their applications. The H-function is defined by the Mellin—Barnes type integral with the integrand containing products and quotients of the Euler gamma functions.
Such a function generalizes most of the known special functions, which means that almost all integral transforms can be put into the form of H-transforms. Another generalization, though a special case of H-transforms, was proposed as the G-transform of the integral transform with the Meijer G-function as a kernel; this generalization includes the classical Laplace and Hankel transforms, the Riemann{Liouville fractional integral transforms, the even and odd Hilbert transforms, the integral transforms with the Gauss hypergeometric function, and others. However, there are transforms that cannot be reduced to a G-transform but can be put into the form of H-transforms: the modified Laplace and Hankel transforms, the Erdelyi—Kober type fractional integration operators, the modified transforms with the Gauss hypergeometric function as kernel, the Bessel type integral transforms, the Mittag-Lefler type integral transforms and others