Multiple Mellin-Barnes integrals in Schwinger-DeWitt technique

We consider off-diagonal asymptotic series for integral kernels of functions of Laplace-type operators on curved backgrounds. These expansions are obtained by applying integral transforms to the DeWitt series for the heat kernel of the corresponding operator and thus represent a DeWitt-type series in the heat kernel coefficients with the coefficients of this expansion (which we call basis kernels) being some hypergeometric-type functions of the Synge world function. Basis kernels of a certain class of operator functions were found previously in terms of $N$-fold Mellin-Barnes integrals. In this paper we study series representations of the corresponding Mellin-Barnes integrals in both non-resonant and resonant cases and suggest a physical interpretation for the emerging series, which is related to the UV and IR properties of operator functions.