THE $ M $TH LEVEL FRACTIONAL DERIVATIVES WITH RESPECT TO ANOTHER FUNCTION

In this paper, we study the integral equation

$ \begin{align*} \varphi( x)=\frac{1}{\Gamma(\alpha)} \int_ a^ x \psi^\prime(t)\Big(\psi( x)-\psi(t)\Big)^{\alpha-1} \phi(t) dt, \quad n-1<\alpha\leq n, 0< a< x< b. \end{align*} $

We establish conditions ensuring the existence of a solution $ \phi( x) $ in the space $ L_1( a, b) $ and derive an explicit formula for this solution. We introduce a novel parametrization of the Hilfer fractional derivative with respect to another function $ \psi $, which unifies and extends several classical operators. In this framework, we develop an extensive variant of Luchko's second level fractional derivative, termed the $ \psi- $second level fractional derivative, encompassing the $ \psi- $Riemann-Liouville, $ \psi- $Caputo, and $ \psi- $Hilfer derivatives as special cases. We analyze the relationships among these derivatives for various parameter values and within different function spaces, discussing their properties and significant results in fractional calculus. Finally, we propose a comprehensive generalization, the $ \psi-m $th level fractional derivative, which facilitates the construction of fractional derivatives of any desired level. The main results focus on the $ \psi- $second level derivative, as it unifies a wide range of established operators, simplifies the interpretation of results, and naturally supports further generalizations.