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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. 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Pronin, W. Wachowski","submitted_at":"2018-12-29T17:05:10Z","abstract_excerpt":"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. 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