Exact and explicit evaluation of Brézin-Hikami kernels

We present exact and explicit form of the kernels Kˆ(x,y) appearing in the theory of energy correlations in the ensembles of Hermitian random matrices with Gaussian probability distribution, see [E. Brézin, S. Hikami, Phys. Rev. E 57 (1998) 4140] and [E. Brézin, S. Hikami, Phys. Rev. E 58 (1998) 7176]. In obtaining this result we have exploited the analogy with the method of producing exact forms of two-sided, symmetric Lévy stable laws, presented by us recently. This result is valid for arbitrary values of parameters in question. We furnish analytical and graphical representations of physical quantities calculated from Kˆ(x,y)ʼs.