A $\varrho$ -Weyl fractional operator of the extended $S$ -type function in a complex domain

This paper proposes a new extension of the well-known $S$ -special function, which is called $\text{SM}$ -function. We introduce this function by drawing inspiration from exponential function. This new special function is studied from a variety of analytical perspectives, including differential and integral operators. Furthermore, the $\varrho$ -Weyl fractional integral operator involving the $\text{SM}$ -function is studied. These classes are defined by utilizing a new q-differential operator.•The $\text{SM}$ - function is provided.•The derivative and integral formulas of the $\text{SM}$ -function are studied.•An application of the $\varrho$ -Weyl fractional integral operator associated with the $\text{SM}$ -function is investigated.