The obrechkoff integral transform: properties and relation to a generalized fractional calculus
- 1 January 2000
- journal article
- research article
- Published by Taylor & Francis Ltd in Numerical Functional Analysis and Optimization
- Vol. 21 (1-2), 121-144
- https://doi.org/10.1080/01630560008816944
Abstract
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[25]. It has been 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],[9],[10] have found operational properties, complex and real inversion formulas, Abel-type theorems for the Obrechkoff transform. This theory has been further developed by Kiryakova [16],[17],[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 and further extensive use of the G-functions. Many other generalized differentiation and integration operators happen to be special cases in this calculus, too. Special cases of the Obrechkoff transform have been "rediscovered" later by many authors. We give examples how their results could be derived from the general ones surveyed here.Keywords
This publication has 3 references indexed in Scilit:
- Convolutional CalculusPublished by Springer Science and Business Media LLC ,1990
- On an integral transformation due to N. ObrechkoffPublished by Springer Science and Business Media LLC ,1980
- Nichtlineare Operatorgleichungen und OperatordifferentialgleichungenMathematische Nachrichten, 1975