α-Mellin Transform and One of Its Applications

MSC 2010: 35R11, 44A10, 44A20, 26A33, 33C45

We consider a generalization of the classical Mellin transformation, called α-Mellin transformation, with an arbitrary (fractional) parameter α > 0. Here we continue the presentation from the paper [5], where we have introduced the definition of the α-Mellin transform and some of its basic properties. Some examples of special cases are provided. Its operational properties as Theorem 1, Theorem 2 (Convolution theorem) and Theorem 3 (α-Mellin transform of fractional R-L derivatives) are presented, and the proofs can be found in [5]. Now we prove some further properties of this integral transform, useful for its application to solving some fractional order differential equations.