Morphing Lord Brouncker's continued fraction for π into the product of Wallis

2011 ◽  
Vol 95 (532) ◽  
pp. 17-22 ◽  
Author(s):  
Thomas J. Osler
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
The Third ◽  
Image Position

Three of the oldest and most celebrated formulae for π are:The first is Vieta's product of nested radicals from 1592 [1]. The second is Wallis's product of rational numbers [2] from 1656 and the third is Lord Brouncker's continued fraction [3,2], also from 1656. (In the remainder of the paper, for continued fractions we will use the more convenient notation

scholarly journals Application of Wynn's epsilon algorithm to periodic continued fractions

1987 ◽  
Vol 30 (2) ◽  
pp. 295-299 ◽  
Author(s):  
M. J. Jamieson
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Quadratic Surd ◽  
Partial Quotients ◽  
Infinite Continued Fraction ◽  
Image Position ◽  
Epsilon Algorithm ◽  
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.

scholarly journals A characterization of rational numbers by p-adic Ruban continued fractions

1985 ◽  
Vol 39 (3) ◽  
pp. 300-305 ◽  
Author(s):  
Vichian Laohakosol
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers

AbstractA type of p–adic continued fraction first considered by A. Ruban is described, and is used to give a characterization of rational numbers.

Positivity and continued fractions from the binomial transformation

2019 ◽  
Vol 149 (03) ◽  
pp. 831-847 ◽  
Author(s):  
Bao-Xuan Zhu
Keyword(s):  
Generating Function ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Hankel Matrix ◽  
Eulerian Polynomials ◽  
Binomial Convolution ◽  
Image Position ◽  
Moment Property

AbstractGiven a sequence of polynomials$\{x_k(q)\}_{k \ges 0}$, define the transformation$$y_n(q) = a^n\sum\limits_{i = 0}^n {\left( \matrix{n \cr i} \right)} b^{n-i}x_i(q)$$for$n\ges 0$. In this paper, we obtain the relation between the Jacobi continued fraction of the ordinary generating function ofyn(q) and that ofxn(q). We also prove that the transformation preservesq-TPr+1(q-TP) property of the Hankel matrix$[x_{i+j}(q)]_{i,j \ges 0}$, in particular forr= 2,3, implying ther-q-log-convexity of the sequence$\{y_n(q)\}_{n\ges 0}$. As applications, we can give the continued fraction expressions of Eulerian polynomials of typesAandB, derangement polynomials typesAandB, general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. In addition, we also prove the strongq-log-convexity of derangement polynomials typeB, Dowling polynomials and Tanny-geometric polynomials and 3-q-log-convexity of general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. We also present a new proof of the result of Pólya and Szegö about the binomial convolution preserving the Stieltjes moment property and a new proof of the result of Zhu and Sun on the binomial transformation preserving strongq-log-convexity.

scholarly journals The evaluation of certain continued fractions

1937 ◽  
Vol 30 ◽  
pp. vi-x
Author(s):  
C. G. Darwin
Keyword(s):  
Difference Equation ◽  
Hypergeometric Function ◽  
Difference Equations ◽  
Continued Fractions ◽  
Continued Fraction ◽  
The Difference ◽  
Image Position

1. If the approximate numerical value of e is expressed as a continued fraction the result isand it was in finding the proof that the sequence extends correctly to infinity that the following work was done. First the continued fraction may be simplified by setting down the difference equations for numerator and denominator as usual, and eliminating two out of every successive three equations. A difference equation is thus formed between the first, fourth, seventh, tenth … convergents , and this equation will generate another continued fraction. After a little rearrangement of the first two members it appears that (1) implies2. We therefore consider the continued fractionwhich includes (2), and also certain continued fractions which were discussed by Prof. Turnbull. He evaluated them without solving the difference equations, and it is the purpose here to show how the difference equations may be solved completely both in his cases and in the different problem of (2). It will appear that the work is connected with certain types of hypergeometric function, but I shall not go into this deeply.

scholarly journals Diophantine approximation by continued fractions

1991 ◽  
Vol 51 (2) ◽  
pp. 324-330 ◽  
Author(s):  
Jingcheng Tong
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Image Position

AbstractLet ξ be an irrational number with simple continued fraction expansion be its ith convergent. Let Mi = [ai+1,…, a1]+ [0; ai+2, ai+3,…]. In this paper we prove that Mn−1 < r and Mn R imply which generalizes a previous result of the author.

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

Ramanujan's Continued Fraction and the Bauer-Muir Transformation

1961 ◽  
Vol 57 (1) ◽  
pp. 76-79 ◽  
Author(s):  
V. N. Singh
Keyword(s):  
Positive Integer ◽  
Gamma Function ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Gamma Functions ◽  
Image Position ◽  
The Right ◽  
Basic Analogue ◽  
Ramanujan's Continued Fraction

Ramanujan's Continued Fraction may be stated as follows: Let where there are eight gamma functions in each product and the ambiguous signs are so chosen that the argument of each gamma function contains one of the specified number of minus signs. Then where the products and the sums on the right range over the numbers α, β, γ, δ, ε: provided that one of the numbers β, γ, δ, ε is equal to ± ±n, where n is a positive integer. In 1935, Watson (3) proved the theorem by induction and also gave a basic analogue. In this paper we give a new proof of Ramanujan's Continued Fraction by using the transformation of Bauer and Muir in the theory of continued fractions (Perron (1), §7;(2), §2).

scholarly journals Limits of unbounded sequences of continued fractions

1990 ◽  
Vol 41 (3) ◽  
pp. 509-512
Author(s):  
Jingcheng Tong
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Partial Quotients ◽  
Limit Points ◽  
Necessary And Sufficient ◽  
Positive Integers

Let X = {xk}k≥1 be a sequence of positive integers. Let Qk = [O;xk,xk−1,…,x1] be the finite continued fraction with partial quotients xi(1 ≤ i ≤ k). Denote the set of the limit points of the sequence {Qk}k≥1 by Λ(X). In this note a necessary and sufficient condition is given for Λ(X) to contain no rational numbers other than zero.

Some properties of the continued fraction expansion of (m/n) e1/q

1970 ◽  
Vol 67 (1) ◽  
pp. 67-74 ◽  
Author(s):  
K. R. Matthews ◽  
R. F. C. Walters
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Positive Integers

Introduction. Continued fractions of the form are called Hurwitzian if b1, …, bh, are positive integers, ƒ1(x), …, ƒk(x) are polynomials with rational coefficients which take positive integral values for x = 0, 1, 2, …, and at least one of the polynomials is not constant. f1(x), …, fk(x) are said to form a quasi-period.

scholarly journals On the Quantitative Metric Theory of Continued Fractions in Positive Characteristic

2018 ◽  
Vol 61 (1) ◽  
pp. 283-293
Author(s):  
Poj Lertchoosakul ◽  
Radhakrishnan Nair
Keyword(s):  
Finite Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Laurent Series ◽  
Positive Characteristic ◽  
Continued Fraction Expansion ◽  
Formal Laurent Series ◽  
Almost Everywhere ◽  
Partial Quotients ◽  
Image Position

AbstractLet 𝔽q be the finite field of q elements. An analogue of the regular continued fraction expansion for an element α in the field of formal Laurent series over 𝔽q is given uniquely by $$\alpha = A_0(\alpha ) + \displaystyle{1 \over {A_1(\alpha ) + \displaystyle{1 \over {A_2(\alpha ) + \ddots }}}},$$ where $(A_{n}(\alpha))_{n=0}^{\infty}$ is a sequence of polynomials with coefficients in 𝔽q such that deg(An(α)) ⩾ 1 for all n ⩾ 1. In this paper, we provide quantitative versions of metrical results regarding averages of partial quotients. A sample result we prove is that, given any ϵ > 0, we have $$\vert A_1(\alpha ) \ldots A_N(\alpha )\vert ^{1/N} = q^{q/(q - 1)} + o(N^{ - 1/2}(\log N)^{3/2 + {\rm \epsilon }})$$ for almost everywhere α with respect to Haar measure.

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