Obrechkoff Integral Transform and Hyper-Bessel Operators via G-function and Fractional Calculus Approach¹

This survey is devoted to one of the most general Laplace-type integral transforms, the so-called Obrechkoff integral transform, introduced and studied for the first time by Obrechkoff, 1958 in [25]. It was modified by Dimovski [5], [6] and used as a basis of a Mikusinski-type operational calculus for the hyper-Bessel differential operators of arbitrary order. Later, in a series of papers Dimovski and Kiryakova [8–10] developed its operational properties and complex and real inversion formulas. This theory was further extended by Kiryakova [15–18] using the tools of the Meijer's G-functions and of the fractional calculus. Namely, a new definition as a G-transform has been given for the Obrechkoff transform. The hyper-Bessel operators themselves, have given rise to a new generalized fractional calculus [17] and further extensive use of the theory of G-functions, [18]. A long list of various generalized differentiation and integration operators happen to be special cases in this calculus. Special cases of the Obrechkoff transform have been “rediscovered” later by many authors. We give some examples.