the exponential functions of central-symmetric $x$-form matrices

it is well known that the matrix exponential function has practical applications in engineering and applied sciences. in this paper, we present some new explicit identities to the exponential functions of a special class of matrices that are known as central-symmetric $x$-form. for instance, $e^{mathbf{a}t}$, $t^{mathbf{a}}$ and $a^{mathbf{a}t}$ will be evaluated by the new formulas in this particular structure. moreover, upper bounds for the explicit relations will be given via subordinate matrix norms. eventually, some numerical illustrations and applications are also adapted.