Integral Transform and Generalized M-Series Fractional Integral Operators (Journal of Mathematical Modeling and Fractional Calculus, 2025)
This paper presents new theorems that build upon existing research by extending the ap plication of the Mellin transform within the framework of fractional integral operators. Traditionally, the Mellin transform has been a powerful tool for analyzing asymptotic behavior, scaling properties, and integral representations of special functions. However, by incorporating fractional integral oper ators, its analytical flexibility is significantly enhanced, allowing for a more in-depth study of special function properties, particularly in fractional calculus. Furthermore, the inclusion of the generalized I-function and M-series broadens this mathematical framework by generalizing established results and encompassing a wider class of special functions. Through their evaluation alongside fractional inte gral operators, this study introduces new integral representations, special cases, and applications that have not been previously explored. This extension greatly increases the applicability of these mathe matical tools in diverse fields such as mathematical physics, engineering, and applied analysis, where fractional calculus and integral transforms play a crucial role in solving complex differential equa tions and boundary value problems. Ultimately, these theorems not only refine existing mathematical structures but also create new opportunities for future research on the interplay between integral trans forms, special functions, and fractional calculus, contributing to both theoretical progress and practical advancements.
Keywords: M-series; I-function; Mellin transform and Fractional Integral