We introduce three families of positive definite bivariate integral kernels whose eigenfunctions and eigenvalues can be explicitly derived in terms of classical orthogonal polynomials. All sequences of classical orthogonal polynomials are involved in at least one of our developments. We show that the well-known Karhunen–Loève expansions of the Wiener, Brownian bridge and Anderson–Darling Gaussian stochastic processes are particular cases of our results.