Inhomogeneous ladder networks,part I

In this paper the theory of orthogonal polynomials is taken as the starting point for developing the theory of semi-infinite inhomogeneous ladder networks of two element kinds. It is shown that the transfer impedance of the ladder network, suitably normalized, is just the Green's function of a self-adjoint difference operator of ‘Sturm–Liouville’ type. An integral representation for this Green's function is given, and is evaluated in product form. The theory is illustrated with a number of examples based on classical orthogonal polynomials, the use of which enables all the calculations to be carried through explicitly.