On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction.

Let the continued fraction expansion of any irrational number be denoted by and let the -th convergent of this continued fraction expansion be denoted by . Let

where . Let . It is shown that if , then the Rogers-Ramanujan continued fraction diverges at . is an uncountable set of measure zero. It is also shown that there is an uncountable set of points such that if , then does not converge generally.

It is further shown that does not converge generally for 1$">. However we show that does converge generally if is a primitive -th root of unity, for some . Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.

     

On the approximation by three consecutive continued fraction convergents

Denote by _pn/_qn,n=1,2,3,…$p_n/q_n,n=1,2,3,\dots$, the sequence of continued fraction convergents of the real irrational number x$x$. Define the sequence of approximation coefficients by _θn:=_qn|_qnx−_pn|,n=1,2,3,…${\theta }_n:=q_n|q_{n}x-p_n|,n=1,2,3,\dots$. A laborious way of determining the mean value of the sequence |_θn+1−_θn−1|,n=2,3,…$|{\theta }_{n+1}-{\theta }_{n-1}|,n=2,3,\dots$, is simplified. The method involved also serves for showing that for almost all x$x$ the pattern _θn−1<_θn<_θn+1${\theta }_{n-1}<{\theta }_n<{\theta }_{n+1}$ occurs with the same asymptotic frequency as the pattern _θn+1<_θn<_θn−1${\theta }_{n+1}<{\theta }_n<{\theta }_{n-1}$, namely 0.12109?$0.12109?$. All the four other patterns have the same asymptotic frequency 0.18945?$0.18945?$. The constants are explicitly given.     

On sums of degrees of the partial quotients in continued fraction expansions of Laurent series

For any formal Laurent series with coefficients cn lying in some given finite field, let x=[a₀(x);a₁(x),a₂(x),…] be its continued fraction expansion. It is known that, with respect to the Haar measure, almost surely, the sum of degrees of partial quotients grows linearly. In this note, we quantify the exceptional sets of points with faster growth orders than linear ones by their Hausdorff dimension, which covers an earlier result by J. Wu.     

An idea on some of Ramanujan’s continued fraction identities

We present an idea on how Ramanujan found some of his beautiful continued fraction identities. Or more to the point: why he chose the ones he wrote down among all possible identities.      

Metrical diophantine approximation for continued fraction like maps of the interval

We study the metrical properties of a class of continued fraction-like mappings of the unit interval, each of which is defined as the fractional part of a Möbius transformation taking the endpoints of the interval to zero and infinity.

     

Dynamics of a Ramanujan-type continued fraction with cyclic coefficients

We study several generalizations of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the continued fraction were determined for certain complex parameters (Borwein et al., Exp. Math. 13, 275–286, 2004, Ramanujan J., in press, 2004; Borwein and Crandall, Exp. Math. 12, 287–296, 2004). A study of the AGM continued fraction is equivalent to an analysis of the convergence of certain difference equations and the stability of dynamical systems. Using the matrix analytical tools developed in 2004, we determine the convergence properties of deterministic difference equations and so divergence of their corresponding continued fractions. Russell Luke’s work was supported in part by a postdoctoral fellowship from the Pacific Institute for the Mathematical Sciences at Simon Fraser University.     

A distributional limit law for the continued fraction digit sum

We consider the continued fraction digits as random variables measured with respect to Lebesgue measure. The logarithmically scaled and normalised fluctuation process of the digit sums converges strongly distributional to a random variable uniformly distributed on the unit interval. For this process normalised linearly we determine a large deviation asymptotic. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Distribution of the reduced quadratic irrationals arising from the odd continued fraction expansion

This paper investigates the quadratic irrationals that arise as periodic points of the Gauss type shift associated to the odd continued fraction expansion. It is shown that these numbers, which we call O-reduced, when ordered by the length of the associated closed primitive geodesic on some modular surface $\Gamma ?\mathbb{H}$, are equidistributed with respect to the Lebesgue absolutely continuous invariant probability measure of the Odd Gauss shift.     

On some generalizations of Ramanujan’s continued fraction identities

In this note we establish continued fraction developments for the ratios of the basic hypergeometric function₂ϕ₁(a,b;c;x) with several of its contiguous functions. We thus generalize and give a unified approach to establishing several continued fraction identities including those of Srinivasa Ramanujan.     

Singular values of the Rogers-Ramanujan continued fraction

Let z∊ C be imaginary quadratic in the upper half plane. Then the Rogers-Ramanujan continued fraction evaluated at q = e ^{2π iz} is contained in a class field of Q(z). Ramanujan showed that for certain values of z, one can write these continued fractions as nested radicals. We use the Shimura reciprocity law to obtain such nested radicals whenever z is imaginary quadratic. 2000 Mathematics Subject Classification Primary—11Y65; Secondary—11Y40     

Simple continued fractions and cutting sequences

Abstract

The connection between cutting sequences of a directed geodesic in the tessellated hyperbolic plane ?², the modular group Γ = PSL(2, ?) and the simple continued fractions of an end point w of the geodesic have been established by Series [13]. In this paper we represent the simple continued fractions of w ∈ ? and the “L” and “R” codes of the cutting sequence in terms of modular and extended modular transformations. We will define a T ₀-path on a graph whose vertices are the set of Farey triangles, as the equivalent of the cutting sequence. The relationship between the directed geodesic with end point w on ?, the Farey tessellation and the simple continued fraction expansion of w ∈ ? _∞ then follows easily as a consequence of this redefinition. Finite, infinite and periodic simple continued fractions are subsequently examined in this light.     

Explicit continued fractions with expected partial quotient growth

For let be the continued fraction expansion of . Write

We construct some numbers 's with

     

On suborbital graphs and related continued fractions

In this paper, we study suborbital graphs for congruence subgroup Γ₀(n) of the modular group Γ to have hyperbolic paths of minimal lengths. It turns out that these graphs give rise to a special continued fraction which is a special case of very famous fraction coming out from Pringsheim’s theorem.     

An algorithm of infinite sums representations and Tasoev continued fractions

For any given real number, its corresponding continued fraction is unique. However, given an arbitrary continued fraction, there has been no general way to identify its corresponding real number. In this paper we shall show a general algorithm from continued fractions to real numbers via infinite sums representations. Using this algorithm, we obtain some new Tasoev continued fractions.

     

A note on bounds for the derivatives of continued fractions

Although it is difficult to differentiate analytic functions defined by continued fractions, it is relatively easy in some cases to determine uniform bounds on such derivatives by perceiving the continued fraction as an infinite composition of linear fractional transformations and applying an infinite chain rule for differentiation.     

Rates of convergence for continued fractions of irrational numbers

The dynamics of the Gauss Map suggests a way to compare the convergence to a real number ζ ε(0,l) of a continued fraction and the divergence of the orbit of ζ Of particular interest is the comparison of the rate of convergence to ζ of its simple continued fraction and the rate of divergence by the Gauss Map of the orbit of ζ for all irrational numbers in (0,l). We state and prove sharp inequalities for the convergence of the sequence of rational convergents of an irrational number ζ. We show that the product of the rate of convergence of the continued fraction of ζ and the rate of divergence by the Gauss Map of the orbit of ζ equals 1.