  Reconstructing a density of states or similar distribution from moments or
continued fractions is an important problem in calculating the electronic and
vibrational structure of defective or non-crystalline solids. For single bands
a quadratic boundary condition introduced previously [Phys. Rev. B 74, 205121
(2006)] produces results which compare favorably with maximum entropy and even
give analytic continuations of Green functions to the unphysical sheet. In this
paper, the previous boundary condition is generalized to an energy-independent
condition for densities with multiple bands separated by gaps. As an example it
is applied to a chain of atoms with s, p, and d bands of different widths with
different gaps between them. The results are compared with maximum entropy for
different levels of approximation. Generalized hypergeometric functions
associated with multiple bands satisfy the new boundary condition exactly.
