Transmutation of MacRobert E-Functions to Fox–Wright Functions via Erdelyi–Kober Fractional Operator

This research will present a comprehensive analysis of the Erdelyi–Kober (E-K) fractional integral operator and its action on various configurations of the hypergeometric form of MacRobert E-function. Analytical solutions for the application of the E-K operator to the simple MacRobert E-function, a weighted form of the function, and the Cauchy product of two distinct E-functions will be derived, resulting in generalized Fox–Wright functions. The study further explores the fundamental mathematical properties, specifically proving the linear property and the closure property within the Fox–Wright domain. Crucially, we will define the convergence criteria required for the validity of these transformations, emphasizing the necessity of the condition Rη>−1 to ensure integral existence at the origin. Furthermore, numerical simulations and graphical comparisons are provided to validate the established analytical identities, demonstrating the stability and asymptotic convergence of the transmuted functional space. Our results will provide a unified framework for evaluating fractional integrals of complex special functions, offering significant utility for problems in mathematical physics and applied analysis involving multiple-parameter functional transmutations.