  The notion of the n-th order local energy, generated by the n-th power of the
Hamiltonian, has been introduced. The n-th order two-particle coalescence
conditions have been derived from the requirements that the n-th order local
energy at the coalescence point is non-singular and equal to the n-th power of
the Hamiltonian eigenvalue. The first condition leads to energy-independent
constraints. The second one is state-specific. The analysis has been done using
a radial, one-dimensional, model Hamiltonian. The model is valid in the
asymptotic region of r ~ 0. The coalescence conditions set the relations
between the expansion coefficients of the radial wave function into a power
series with respect to r.
