Continued fraction as a discrete nonlinear transform

  title={Continued fraction as a discrete nonlinear transform},
  author={Carl M Bender and Kimball A. Milton},
  journal={Journal of Mathematical Physics},
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}. This simplification is illustrated in the context of graph combinatorics. 
6 Citations

Quasi‐exactly solvable systems and orthogonal polynomials

This paper shows that there is a correspondence between quasi‐exactly solvable models in quantum mechanics and sets of orthogonal polynomials {Pn}. The quantum‐mechanical wave function is the

A class of exactly-solvable eigenvalue problems

The class of differential equation eigenvalue problems −y''(x) + x2N+2y(x) = xN Ey(x) (N = −1, 0, 1, 2, 3, ...) on the interval −∞ < x < ∞ can be solved in closed form for all the eigenvalues E and

Spherically symmetric random walks in noninteger dimension

A previous article proposed a new kind of random walk on a spherically symmetric lattice in arbitrary noninteger dimension D. Such a lattice avoids the problems associated with a hypercubic lattice

The Caldeira?Leggett quantum master equation in Wigner phase space: continued-fraction solution and application to Brownian motion in periodic potentials

The continued-fraction method of solving classical Fokker?Planck equations has been adapted to tackle quantum master equations of the Caldeira?Leggett type. This was done taking advantage of the

Solving spin quantum master equations with matrix continued-fraction methods: application to superparamagnets

We implement continued-fraction techniques to solve exactly quantum master equations for a spin with arbitrary S coupled to a (bosonic) thermal bath. The full spin density matrix is obtained, so that



Asymptotic graph counting techniques in ψ2N field theory

We discuss two different techniques for obtaining asymptotic estimates of the number of n‐vertex graphs in a ψ2N field theory as n→∞. The first technique relies on difference equations and the second

The planar approximation. II

The planar approximation is reconsidered. It is shown that a saddle point method is ineffective, due to the large number of degrees of freedom. The problem of eliminating angular variables is

J. Math. Phys

  • J. Math. Phys
  • 1978

3 For the results on Euler and Bernoulli numbers see Ref. 2

  • 3 For the results on Euler and Bernoulli numbers see Ref. 2
  • 1987

Analytic Theory of Continued Fractions (Van Nostrand

  • New York,
  • 1948