Mikusiński's Operational Calculus for the Transmuted 1st‐Level General Fractional Derivatives

ABSTRACT The transmuted ‐fold sequential 1st‐level general fractional derivatives are introduced within the framework of conjugation (transmutation) and Sonin kernels, providing a unified extension of general fractional derivatives that includes the Riemann–Liouville and Caputo types, as well as their transmuted versions, as particular cases. A Mikusiński‐type operational calculus is developed for both the transmuted 1st‐level general fractional derivatives and the transmuted ‐fold sequential 1st‐level general fractional derivatives, in which these operators are interpreted as algebraic elements in the field of convolution quotients. This approach enables the reduction of fractional differential equations to algebraic equations and yields explicit solution formulas for initial‐value problems, including linear multi‐term fractional differential equations with constant coefficients. The fundamental properties of the proposed operators are established, in particular, the validity of the first and second fundamental theorems of generalized fractional calculus. The developed framework generalizes several known operational calculi and provides a systematic method for transferring results from the classical setting to more general fractional models. As special cases, the classical results for the Riemann–Liouville and Caputo fractional derivatives, along with their transmuted versions, are recovered.