Cubic string boundary value problems and Cauchy biorthogonal polynomials

Cauchy biorthogonal polynomials appear in the study of special solutions to the dispersive nonlinear partial differential equation called the Degasperis–Procesi (DP) equation, as well as in certain two-matrix random matrix models. Another context in which such biorthogonal polynomials play a role is the cubic string; a third-order ODE boundary value problem –f ''' = zgf which is a generalization of the inhomogeneous string problem studied by Krein. A general class of such boundary value problems going beyond the original cubic string problem associated with the DP equation is discussed under the assumption that the source of inhomogeneity g is a discrete measure. It is shown that by a suitable choice of a generalized Fourier transform associated with these boundary value problems one can establish a Parseval type identity which aligns Cauchy biorthogonal polynomials with certain natural orthogonal systems on L2g.