This article extends new left-sided ( \(\mathtt{L.S}\) ) fractional derivatives and integrals to higher orders with respect to another ( \(\mathtt{wrta}\) ) function \(\varphi\) , which involves a weighted function \(\omega\) and the modified generalized Mittag-Leffler function ( \(\mathtt{GMLF}\) ) in the context of Riemann-Liouville ( \(\mathtt{RL}\) ) and Caputo operators. Additionally, several operators of non-singular kernels are deduced as special cases of the study. Also, some important properties of the proposed operators are proved. Building on the new general \(Q\) -operator of a function \(\mathtt{wrta}\) function \(\varphi\) , we introduce its reciprocal operator and present the general fundamental relations between the left and right ( \(\mathtt{L\&R}\) ) sides to any \((\omega,\varphi)\) -fractional integrals and derivatives. As a consequence, we have derived the right-sided ( \(\mathtt{R.S}\) ) definitions of our proposed operators, along with their characteristics. Furthermore, fixed point theorems ( \(\mathtt{FPTs}\) ) of the Banach and Schaefer are utilized to examine the qualitative results of solutions for \((\omega,\varphi)\) -Caputo fractional integro-differential equations ( \(\mathtt{IDEs}\) ). Finally, two mathematical applications are demonstrated to confirm the effectiveness of our outcomes.
