Saigo fractional integral operators with incomplete R -functions kernel

Fractional calculus has emerged as a robust analytical framework with wide-ranging applications across mathematical analysis, including differential and integral equations, special functions, and series representations. Motivated by these developments, the present study focuses on analyzing and extending Saigo-type fractional integral operators. Building upon this foundation, we introduce a Saigo fractional integral operator characterized by an incomplete R-function kernel. Owing to the general structure of the proposed operator and the associated special functions, several known and new results are obtained as particular cases. These include image formulas involving the Riemann–Liouville and Erdélyi–Kober fractional integral operators, as well as representations connected with generalized Fox H-function, Wright hypergeometric, Mittag–Leffler, and Whittaker functions. The derived image formulas provide a unified framework for studying a broad class of special functions and may be useful in applications of fractional differential and integral equations. Furthermore, several earlier results reported in the literature arise naturally as special cases of the present findings, demonstrating the unifying and general nature of the proposed approach.