General Transmutation Relations and Their Applications

We consider a very general class of fractional calculus operators, given by transmuting the classical fractional calculus along an arbitrary invertible linear operator S. Specific cases of S, such as shift, reflection, and composition operators, give rise to well-known settings such as that of fractional calculus with respect to functions, and allow simple connections between left-sided and right-sided fractional calculus with different constants of differintegration. We define, for the first time, general transmuted versions of the Laplace transform and convolution of functions, and discuss how these ideas can be used to solve fractional differential equations in more general settings. Copyright (C) 2024 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)