On analogues of exponential functions for antisymmetric fractional derivatives

An equation with the antisymmetric fractional derivative of order α∈(1,2)α∈(1,2), containing the tβtβ-potential is solved using the Mellin transform method. The solutions are analogues of exponential functions of a new type. They are represented as Meijer G-function series in a finite time interval. In the classical limit α⟶1+α⟶1+, the eigenfunction equation for a derivative of the first order and its solution–an exponential function, are recovered. Then an analogy between the derivation of Euler–Lagrange equations in fractional mechanics and in classical mechanics is discussed. The results are applied to a simple fractional Euler–Lagrange equation containing an antisymmetric fractional derivative and its general solution is obtained.