Heat kernel for higher-order differential operators in Euclidean space

We consider heat kernel for higher-order operators with constant coefficients in $d$-dimensio\-nal Euclidean space and its asymptotic behavior. For arbitrary operators which are invariant with respect to $O(d)$-rotations we obtain exact analytical expressions for the heat kernel and Green functions in the form of infinite series in Fox--Wright psi functions and Fox $H$-functions. We investigate integro-differential relations and the asymptotic behavior of the functions $\mathcal{E}_{\nu, \alpha}(z)$, in terms of which the heat kernel of $O(d)$-invariant operators are expressed. It is shown that the obtained expressions are well defined for non-integer values of space dimension $d$, as well as for operators of non-integer order. Possible applications of the obtained results in quantum field theory and the connection with fractional calculus are discussed.