The asymptotic behavior of a dynamical process coming from the development in continuous fractions

The aim of this paper is the study of a dynamical process generated by a sequence of maps: $${x_{n + 1}} = {f_n}\left( {{x_n}} \right)$$ where $${f_n}{\rm{ : }}\left( {0,\infty } \right){\rm{ }} \to \left( {0,\infty } \right){\rm{, }}{f_n}{\rm{ }}\left( x \right){\rm{ = }}{{{c_n}} \over {1 + x}}{\rm{ for all }}n{\rm{ }} \in {\rm{ }}N{\rm{ and }}{\left( {{c_n}} \right)_n}$$ is a a sequence of positive numbers. This process is generated similar to continuous fractions development. A continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the inverse of another number, then writing this other number as the sum of its integer part and another inverse, and so on. In a finite continued fraction (or terminated continued fraction), the iteration is terminated after finitely many steps by using an integer in stead of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers are called the coefficients or terms of the continued fraction. We will study the pre-equilibrium points for this process, the attraction basins and the stability.

Chebyshev Polynomials and Continued Fractions Related

Let $p$, $q$ be complex polynomials, $\deg p>\deg q\geq 0$. We consider the family of polynomials defined by the recurrence $P_{n+1}=2pP_n-qP_{n-1}$ for $n=1, 2, 3, ...$ with arbitrary $P_1$ and $P_0$ as well as the domain of the convergence of the infinite continued fraction $$f(z)=2p(z)-\cfrac{q(z)}{2p(z)-\cfrac{q(z)}{2p(z)-...}}$$ null

A simple proof of Euler's continued fraction of e1/M

A continued fraction is an expression of the formand we will denote it by the notation [f0, (g0, f1), (g1, f2), (g2, f3), … ]. If the numerators gi are all equal to 1 then we will use a shorter notation [f0, f1, f2, f3, … ]. A simple continued fraction is a continued fraction with all the gi coefficients equal to 1 and with all the fi coefficients positive integers except perhaps f0.The finite continued fraction [f0, (g0, f1), (g1, f2),…, (gk–1, fk)] is called the k th convergent of the infinite continued fraction [f0, (g0, f1), (g1, f2),…]. We defineif this limit exists and in this case we say that the infinite continued fraction converges.

A COUNTEREXAMPLE TO A CONTINUED FRACTION CONJECTURE

It is known that if $a\in\mathbb{C}\setminus(-\infty,-\tfrac14]$ and $a_n\to a$ as $n\to\infty$, then the infinite continued fraction with coefficients $a_1,a_2,\dots$ converges. A conjecture has been recorded by Jacobsen et al., taken from the unorganized portions of Ramanujan’s notebooks, that if $a\in(-\infty,-\tfrac14)$ and $a_n\to a$ as $n\to\infty$, then the continued fraction diverges. Counterexamples to this conjecture for each value of $a$ in $(-\infty,-\tfrac14)$ are provided. Such counterexamples have already been constructed by Glutsyuk, but the examples given here are significantly shorter and simpler.

The relaxation function of the spin van der Waals model in a weak magnetic field

We consider the spin van der Waals model in a weak magnetic field. For this model we obtain the relaxation function by applying the perturbative method that was developed for the evaluation of the relaxation function represented as an infinite continued fraction. To show the applicability of the perturbative method, we make comparisons between an exact result and various degrees of approximation.

Dynamical perturbation theory for the evaluation of the relaxation function

Use of the infinite continued-fraction representation has played an important role in the evaluation of the relaxation function of many-body systems. By using the method of recurrence relations, we propose here a dynamical perturbation theory for the evaluation of the infinite continued fraction. The analytic expression of the relaxation function is given in terms of the known relaxation function for the unperturbed system. Two illustrative examples are also given.

Application of Wynn's epsilon algorithm to periodic continued fractions

The infinite continued fractionin whichis periodic with period l and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational numbers [1]. We shall also assume that they are positive. The transformation discussed below applies only to pure periodic fractions where n is zero.