Related papers: Some conjectural continued fractions
We prove a continued fraction expansion for the reciprocal of a certain $q$-series. All the specialists in the world are asked whether it is new or not.
We give combinatorial descriptions of the terms occurring in continuants of general continued fractions that diverge to three limits. Equating these with the usual combinatorial descriptions due to Euler, Sylvester, and Minding induces…
Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their…
In this paper we present many congruences for several Ap\'ery-like sequences.
We give a concise introduction to the theory of continuants and show how Perron used them in his proof of Tietze theorem on the convergence of infinite semi-regular continued fractions, as well as for the study of the convergence of purely…
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…
The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…
We derive closed-form expressions for several new classes of Hurwitzian- and Tasoevian continued fractions, including $[0;\overline{p-1,1,u(a+2nb)-1,p-1,1,v(a+(2n+1)b)-1 }\,\,]_{n=0}^\infty$, $[0; \overline{c + d m^{n}}]_{n=1}^{\infty}$ and…
In this paper we study the integrals of fractional parts of given functions, and develop some new tools to understand the behaviour of prime differences. We demonstrate how simply some seemingly difficult conjectures related to prime…
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.
We introduce here a general framework for studying continued fraction expansions for complex numbers and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial…
By some new recursive algorithms, in this paper, we will give some improvements on Waring's problem.
In this short note we report on results on a computational search for a counterexample to the strong coincidence conjecture. In particular, we discuss the method used so that further searches can be conducted.
In this paper, we introduce telescoping continued fractions to find lower bounds for the error term $r_n$ in Stirling's approximation $\displaystyle n! = \sqrt{2\pi}n^{n+1/2}e^{-n}e^{r_n}.$ This improves lower bounds given earlier by…
The classical continued fraction is generalized for studying the rational approximation problem on multi-formal Laurent series in this paper, the construction is called m-continued fraction. It is proved that the approximants of an…
The algorithm of Shor for prime factorization is a hybrid algorithm consisting of a quantum part and a classical part. The main focus of the classical part is a continued fraction analysis. The presentation of this is often short, pointing…
We present an approach for regression problems that employs analytic continued fractions as a novel representation. Comparative computational results using a memetic algorithm are reported in this work. Our experiments included fifteen…
We give an extensive list of parametrized families of polynomial continued fractions of smallest possible degrees for $\pi^2$ and $\zeta(3)$, and mention similar results for other constants.
We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers.
Some new integrals involving the Stieltjes constants are developed in this paper.