  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.
