A -Weyl fractional operator of the extended -type function in a complex domain.

This paper proposes a new extension of the well-known -special function, which is called -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 -Weyl fractional integral operator involving the -function is studied. These classes are defined by utilizing a new q-differential operator.•The - function is provided.•The derivative and integral formulas of the -function are studied.•An application of the -Weyl fractional integral operator associated with the -function is investigated.