相关论文: Ap\'ery Acceleration of Continued Fractions
Following Apery's proof of the irrationality of zeta(3), Beukers found an elegant reinterpretation of Apery's arguments using modular forms. We show how Beukers arguments can be adapted to a p-adic setting. In this context, certain…
We propose and study a generalized continued fraction algorithm that can be executed in an arbitrary imaginary quadratic field, the novelty being a non-restriction to the five Euclidean cases. Many hallmark properties of classical continued…
Continued fractions in the field of $p$--adic numbers have been recently studied by several authors. It is known that the real continued fraction of a positive quadratic irrational is eventually periodic (Lagrange's Theorem). It is still…
By applying the MC algorithm and the Bauer-Muir transformation for continued fractions, in this paper we shall give six examples to show how to establish an infinite set of continued fraction formulas for certain Ramanujan-type series, such…
In the course of the proof of the irrationality of zeta(2) R. Apery introduced numbers b_n = \sum_{k=0}^n {n \choose k}^2{n+k \choose k}. Stienstra and Beukers showed that for the prime p > 3 Apery numbers satisfy congruence b((p-1)/2) =…
By a classical result of Gauss and Kuzmin, the continued fraction expansion of a ``random'' real number contains each digit $a\in\mathbb{N}$ with asymptotic frequency $\log_2(1+1/(a(a+2)))$. We generalize this result in two directions:…
In a recent work on Euler-type formulae for even Dirichlet beta values, i.e. $\beta{(2n)}$, I have derived an exact closed-form expression for a class of zeta series. From this result, I have conjectured closed-form summations for two…
Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we…
If a continued fraction $K_{n=1}^{\infty} a_{n}/b_{n}$ is known to converge but its limit is not easy to determine, it may be easier to use an extension of $K_{n=1}^{\infty}a_{n}/b_{n}$ to find the limit. By an extension of…
Continued fractions have been introduced in the field of $p$--adic numbers $\mathbb{Q}_p$ by several authors. However, a standard definition is still missing since all the proposed algorithms are not able to replicate all the properties of…
In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of…
Casually introduced thirty years ago, a simple algebraic equation of degree 4, with coefficients in Fp[T], has a solution in the field of power series in 1/T, over the finite field Fp. For each prime p > 3, the continued fraction expansion…
Via the MC-algorithm, in this paper we produce seven continued fraction formulae involving products and quotients of three gamma functions with three parameters, and another is an extension of Entry 34 in Chapter 12 of Ramanujan's second…
A classical theorem in continued fractions due to Serret shows that for any two irrational numbers x and y related by a transformation $\gamma$ in PGL(2,Z) there exist s and t for which the complete quotients x_s and y_t coincide. In this…
We prove the exponential growth of the cardinality of the set of numbers of spanning trees in simple (and planar) graphs on $n$ vertices, answering a question of Sedl\'a\v{c}ek from 1969. The proof uses a connection with continued…
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…
We study explicit continued fraction expansions for certain series. Some of these expansions have symmetry that generalizes some remarkable examples discovered independently by Kmosek and Shallit. Furthermore, we prove the following…
In this paper we study the properties of an algorithm for generating continued fractions in the field of p-adic numbers $\mathbb{Q}_p$. First of all, we obtain an analogue of the Galois' Theorem for classical continued fractions. Then, we…
In a recent paper G. Bhatnagar has given simple proofs of some of Ramanujan's continued fractions. In this note we show that some variants of these continued fractions are generating functions of q-Schroeder-like numbers.
Continued fractions have a long history in number theory, especially in the area of Diophantine approximation. The aim of this expository paper is to survey the main results on the theory of $p$--adic continued fractions, i.e. continued…