On Some Generalizations of the Bi-Ordinal Hilfer’s Fractional Derivative

The author considers a generalization of the concept of bi-ordinal Hilfer’s fractional derivative, which he introduced earlier. In particular, the concept of bi-ordinal Hilfer’s derivative of a function with respect to another function and proportional bi-ordinal Hilfer’s derivative of a function with respect to another function are introduced. The main compositional properties for operators of bi-ordinal fractional derivatives and integrals are given. A formula for the Laplace transform of the proportional bi-ordinal Hilfer derivative is obtained and closed-form solutions to the Cauchy-type problems for linear equations with the mentioned generalized bi-ordinal Hilfer’s fractional derivatives are constructed.