In this paper, we propose some fractional integral identities between the \({}_pR_q(\lambda ,\eta ; z)\) function and Riemann–Liouville fractional integral operators. In particular, we derive explicit integral expressions of the \({}_pR_q(\lambda ,\eta ; z)\) function in terms of Riemann–Liouville fractional integral operators associated with generalized M-series. Furthermore, an analytical solution to the free electron laser equation has also been discussed. A closed-form solution is derived in terms of the \({}_pR_q(\lambda ,\eta ;z)\) function by using Riemann–Liouville fractional operators.
