Spectral residues of second-order differential equations: A new method forsummation identities and inversion formulas

This article deals with differential equations with spectral parameter from the point of view of formal power series. The treatment does not make use of the notion of eigenvalue, but introduces a new idea: the spectral residu e. The article focuses on second-order, self-adjoint problems. In such a setti ng, every potential function determines a sequence of spectral residues. Th is correspondence is invertible and gives rise to a combinatorial inversion formula. Other interesting combinatorial consequences are obtained by cons idering spectral residues of exactly solvable potentials of one-dimensional quantum mechanics. It is also shown that the Darboux transformation of one-dimensional potenti als corresponds to a simple negation of the corresponding spectral residues . This fact leads to another combinatorial inversion formula. Finally, ther e is a brief discussion of applications. The topics considered are enumerat ion problems and integrable systems.