In this article, we aim to develop new k , s -fractional integral and differential operators containing S -functions as kernels in a form of generalized k -Mittag-Leffer functions. We also set up various properties of such operators. Furthermore, we consider a variety of implications of the major outcomes that will be very useful in the implementation of scientific, engineering, and technical problems.

In this paper, we present two new unified integral formulas involving a generalized V-function. Some interesting special cases of the main results are also considered in the form of corollaries. Due to the general nature of the V-function, several results involving different special functions such as the exponential function, the Mittag-Leffler function, the Lommel function, the Struve function, the Wright generalized Bessel function, the Bessel function and the generalized hypergeometric function are obtained by specializing the parameters in the presented formulas. More results are also discussed in detail.

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In the present paper, we establish some composition formulas for Marichev-Saigo-Maeda (MSM) fractional calculus operators with V -function as the kernel. In addition, on account of V -function, a variety of known results associated with special functions such as the Mittag-Leffler function, exponential function, Struve’s function, Lommel’s function, the Bessel function, Wright’s generalized Bessel function, and the generalized hypergeometric function have been discovered by defining suitable values for the parameters.

In this study, we form integral formulas for Saigo’s hypergeometric integral operator involving V-function. Corresponding assertions for the classical Riemann–Liouville (R-L) and Erdélyi–Kober (E-K) fractional integral operator are extrapolated. Also, by putting in the transformations of Beta and Laplace, we can establish their composition formulas. By selecting the appropriate parameter values, the V-function may be reduced to a variety of functions, including the exponential function, Mittag–Leffler, Lommel, Struve, Wright’s generalized Bessel function, and Bessel and generalized hypergeometric function.