Related papers: Continued fraction expansion for function sec(x) +…
Elementary problems like the evaluation of repeated derivatives of ordinary transcendent functions can usefully be treated by the use of special polynomials and of a formalism borrowed from combinatorial analysis. Motivated by previous…
The direct calculation of the Generalized operator entropy proves difficult by the appearance of rational exponents of matrices. The main motivation of this work is to overcome these difficulties and to present a practical and efficient…
A fractal function is a function whose graph is the attractor of an iterated function system. This paper generalizes analytic continuation of an analytic function to continuation of a fractal function.
In this short note we prove two elegant generalized continued fraction formulae $$e= 2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\ddots}}}}$$ and $$e= 3+\cfrac{-1}{4+\cfrac{-2}{5+\cfrac{-3}{6+\cfrac{-4}{7+\ddots}}}}$$ using elementary…
In this paper Euler shows how, if we have recursive functions f,g,h and an infinite sequence A,B,C,... which satisfies fA=gB+hC, f'B=g'C+h'D, f''C=g''D+h''E, f'''D=g'''E+h'''F, etc., where the primes denote an index not a derivative, then…
The author proposed an elementary derivation of the formula of Yu.V. Nesterenko expansion in continued fraction of a number 2*$\zeta(3)$.
We derive continued fractions for partition generating functions, utilizing both Euler's techniques and Ramanujan's techniques. Although our results are for integer partitions there is scope to extend this work to vector partitions,…
In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…
A new series expansion for the function (arctan(x)/x)^n is developed and some properties of the expansion coefficients are obtained.
We introduce a random dynamical system related to continued fraction expansions. It uses random combination of the Gauss map and the R\'enyi (or backwards) continued fraction map. We explore the continued fraction expansions that this…
We consider series of the form $$ \frac{p}{q} +\sum_{j=2}^\infty \frac{1}{x_j}, $$ where $x_1=q$ and the integer sequence $(x_n)$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n|x_{n+1}$ for $n\geq 1$.…
We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new…
The aim of the present paper is to give extensions of the cosine-sine functional equation.
In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions with deterministic terms are…
A well known method for convergence acceleration of continued fraction $\K(a_n/b_n)$ is to use the modified approximants $S_n(\omega_n)$ in place of the classical approximants $S_n(0)$, where $\omega_n$ are close to tails $f^{(n)}$ of…
This note presents an especially short and direct variant of Hermite's proof of the simple continued fraction expansion e = [2,1,2,1,1,4,1,1,6,...] and explains some of the motivation behind it.
New expansions of the number zeta(3) in continuous fractions are found.
We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Numerous distinct such expansions are possible for a complex number. They can be arrived at through various algorithms, as…
The functional interpolation problem on a continual set of nodes by an integral continued C-fraction is studied. The necessary and sufficient conditions for its solvability are found. As a particular case, the considered integral continued…
We state and prove three general formulas allowing to transform formal finite sums into formal continued fractions and apply them to generalize certain expansions in continued fractions given by Hone and Varona.