# Heat kernel for higher-order differential operators in Euclidean space

## TL;DR

This paper derives exact analytical expressions for the heat kernel of higher-order differential operators in Euclidean space, including non-integer dimensions and orders, with applications in quantum field theory and fractional calculus.

## Contribution

It provides explicit formulas for heat kernels of $O(d)$-invariant higher-order operators using Fox functions, extending to non-integer dimensions and orders.

## Key findings

- Exact formulas for heat kernels and Green functions in terms of Fox functions.
- Asymptotic analysis of special functions related to the heat kernel.
- Applicability to non-integer dimensions and operator orders.

## Abstract

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.