Fractional differential operators in the complex matrix-variate case

There are many applications for fractional integrals, fractional derivatives and fractional differential equations in the real scalar variable cases. However, when it comes to fractional integrals in the real and complex matrix-variate cases there are not many papers. The author has given some definitions and properties for fractional integrals and fractional derivatives in the real matrix-variate cases recently. Some situations of fractional derivatives in the complex matrix-variate case are discussed in the present article, along with the definitions for fractional derivatives in the complex matrix-variate case. The definition introduced here enables us to handle certain types of derivatives of real-valued scalar functions of matrix argument in the complex domain. It is not universally applicable. Fractional derivatives in the Riemann–Liouville and Caputo senses are evaluated when the arbitrary function is compatible with right and left sided fractional integrals in the complex matrix-variate cases. Kober operators of the first and second kind in the complex matrix-variate cases are also examined here.