相关论文: Ap\'ery Acceleration of Continued Fractions
We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections…
In this paper, we represent a continued fraction expression of Mathieu series by a continued fraction formula of Ramanujan. As application, we obtain some new bounds for Mathieu series.
The aim of this paper is to study the convergence and divergence of the Rogers-Ramanujan and the generalized Rogers-Ramanujan continued fractions on the unit circle. We provide an example of an uncountable set of measure zero on which the…
We give a direct and simple proof of Touchard's continued fraction, provide an extension of it, and transform it into similar expansions related to Motzkin and Schroeder numbers. Another proof is then given that uses only induction. We use…
In this paper, we will first summarize known results concerning continued fractions. Then we will limit our consideration to continued fractions of quadratic numbers. The second author described periods and sometimes precise form of…
The Ramanujan Machine project detects new expressions related to constants of interest, such as $\zeta$ function values, $\gamma$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures concerning the…
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…
Let $p$ be a prime number and $K$ be a field with embeddings into $\mathbb{R}$ and $\mathbb{Q}_p$. We propose an algorithm that generates continued fraction expansions converging in $\mathbb{Q}_p$ and is expected to simultaneously converge…
We revisit Beukers' modular-form proof of the irrationality of $\zeta(3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group $\Gamma_0(6)^\star$, we show that Beukers' choice is not isolated: it belongs to…
We prove a suite of dynamical results, including exactness of the transformation and piecewise-analyticity of the invariant measure, for a family of continued fraction systems, including specific examples over reals, complex numbers,…
We present a new `elementary' proof of the irrationality of $\zeta(3)$ based on some recent `hypergeometric' ideas of Yu.Nesterenko, T.Rivoal, and K.Ball, and on Zeilberger's algorithm of creative telescoping.
We present here continued fraction for Zeta(3) parametrized by some family of points (F,G) on projective line. This family of points can be obtained if from full projective line would be removed some no more than countable exeptional set of…
In 1985, Robbins observed by computer the continued fraction expansion of certain algebraic power series over a finite field. Incidentally, he came across a particular equation of degree 4 in characteristic p=13. This equation has an…
The Ramanujan Machine project predicts new continued fraction representations of numbers expressed by important mathematical constants. Generally, the value of a continued fraction is found by reducing it to a second order linear difference…
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.
In \cite{d4}, we gave a method to construct a continued fraction of the function $F(x):=e^{x}E_{1}(x)$. More precisely we define $F_{1}(x)$ as the reciprocal of $F(x)$ and we inductively define $F_{m}(x)$ as the reciprocal of ``$F_{m-1}(x)$…
Using WZ pairs we present an infinite family of accelerated series for computing $\zeta(3)$.
Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for zeta(4n+3) which generalizes Apery's series for zeta(3), and appears to give the best possible series…
The inspiral of two compact objects in gravitational wave astronomy is described by a post-Newtonian expansion in powers of $(v/c)$. In most cases, it is believed that the post-Newtonian expansion is asymptotically divergent. A standard…
We prove bounds for the popularity of products of sets with weak additive structure, and use these bounds to prove results about continued fractions. Namely, we obtain a nearly sharp upper bound for the cardinality of Zaremba's set modulo…