The quantization of gauge fields and gravitation on manifolds with boundary makes it necessary to study boundary conditions which involve both normal and tangential derivatives of the quantized field. The resulting 1-loop divergences can be studied by means of the asymptotic expansion of the heat kernel, and a particular case of their general structure is analysed here in detail. The interior and boundary contributions to heat-kernel coefficients are written as linear combinations of all geometric invariants of the problem. The behaviour of the differential operator and of the heat kernel under conformal rescalings of the background metric leads to recurrence relations which, jointly with the boundary conditions, may determine these linear combinations. Remarkably, they are expressed in terms of universal functions, independent of the dimension of the background and invariant under conformal rescalings, and new geometric invariants contribute to heat-kernel asymptotics. Such a technique is applied to the evaluation of the coefficient when the matrices occurring in the boundary operator commute with each other. Under these assumptions, the form of the and coefficients is obtained for the first time, and new equations among universal functions are derived. A generalized formula, relating asymptotic heat kernels with different boundary conditions, is also obtained.