Continued Fraction as a Discrete Nonlinear Transform

The connection between a Taylor series and a continued-fraction involves a nonlinear relation between the Taylor coefficients {an} and the continued-fraction coefficients {bn}. In many instances it turns out that this nonlinear relation transforms a complicated sequence {an} into a very simple one {bn}. We illustrate this simplification in the context of graph combinatorics. PACS numbers: 02.90.+p, 11.90.+t, 11.10.-z 1 The purpose of a transform is to convert an apparently complicated problem into one that is obviously simple. In order to be useful a transform must have an inverse. One applies the transform to a difficult-looking problem, solves the resulting easy problem, and applies the inverse transform to obtain the solution to the original problem. A typical example of a linear transform is the Fourier transform: F [f ] ≡ 1 √ 2π ∫ ∞ −∞ dx ef(x) = F (y). (1a) The inverse transform is defined by: F[F ] ≡ 1 √ 2π ∫ ∞ −∞ dy eF (y) = f(x). (1b) The Fourier transform (1a) converts the heat equation ut = uxx, which is a partial differential equation for u(x, t), into the ordinary differential equation Ut = −yU for the function U(y, t). This ordinary differential equation is easy to solve and one need only apply the inverse Fourier transform (1b) to the solution of the ordinary differential equation to obtain the solution to the original heat equation. One solves the Korteweg-deVries equation, a difficult nonlinear wave equation, by means of an interesting nonlinear transform, which converts the original partial differential equation into a simple linear problem involving isospectral flow. The inverse transform is performed by the method of inverse scattering. In this paper we investigate a nonlinear transform that converts the discrete sequence a1, a2, a3, . . . into another sequence b1, b2, b3, . . .. The first four equations for this transform are: b1 = a1, (2a) b2 = −a1 + a2/a1, (2b) b3 = a1a3 − a2 a1a2 − a1 , (2c) b4 = a1a2a4 − a1a4 − a1a3 + 2a1a2a3 − a1a2 a1a2a3 − a1a3 − a2 + a1a2 . (2d) The first four equations for the inverse transform are: a1 = b1, (3a) a2 = b1(b1 + b2), (3b) 2 a3 = b1[b2b3 + (b1 + b2) ], (3c) a4 = b1[b2b3(b3 + b4) + 2(b1 + b2)b2b3 + (b1 + b2) ]. (3d) Observe that the structure of this transform is triangular in the sense that the first n terms in the a sequence uniquely determine bn and the first n terms in the b sequence uniquely determine an. We can derive the formulas in (2) and (3) for this transformation very simply. Consider the formal Taylor series 1 + ∞