  We establish a new natural extension of Mittag-Leffler function with three
variables which is so called "trivariate Mittag-Leffler function". The
trivariate Mittag-Leffler function can be expressed via complex integral
representation by putting to use of the eminent Hankel's integral. We also
investigate Laplace integral relation and convolution result for a univariate
version of this function. Moreover, we present fractional derivative of
trivariate Mittag-Leffler function in Caputo type and we also discuss Riemann-
Liouville type fractional integral and derivative of this function. The link of
trivariate Mittag-Leffler function with fractional differential equation
systems involving different fractional orders is necessary on certain
applications in physics. Thus, we provide an exact analytic solutions of
homogeneous and inhomogeneous multi-term fractional differential equations by
means of a newly defined trivariate Mittag-Leffler functions.
