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.      

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.     

Some asymptotic results on digits of the nearest integer continued fraction

By using stochastic dependence with complete connections we obtain some asymptotic formulas on the digits of the nearest integer continued fraction. We get a formula for the geometric mean of partial denominators and the relative frequency of a pair (α, ϵ) of digits, of numerators and of signs.     

Some general theorems on the explicit evaluations of Ramanujan''s cubic continued fraction

In 2001, Jinhee Yi found many explicit values of the famous Rogers–Ramanujan continued fraction by using modular equations and transformation formulas for theta-functions. In this paper, we use her method to find some general theorems for the explicit evaluations of Ramanujan's cubic continued fraction.     

An entry of Ramanujan on continued fraction involving the gamma function in his notebooks

In this paper we give a new proof of Ramanujan?s continued fraction involving the Gamma function, Entry 39 of Chapter 12 of Ramanujan?s second notebook, by using Watson?s form of the Bauer–Muir transformation.     

Some distribution results of the Oppenheim continued fractions

We study some distribution properties of the Oppenheim continued fraction expansions. A Gauss–Kuzmin–Lévy type theorem is established. Based on this, a Fréchet law concerning the partial maxima of the growth rate of the digit sequence is derived, which extends previous work of Galambos and Philipp on the regular continued fraction expansion. Besides, a uniform distribution modulo 1 result is obtained.     

Some theta function identities related to the Rogers-Ramanujan continued fraction

In his first and second letters to Hardy, Ramanujan made several assertions about the Rogers-Ramanujan continued fraction . In order to prove some of these claims, G. N. Watson established two important theorems about that he found in Ramanujan's notebooks. In his lost notebook, after stating a version of the quintuple product identity, Ramanujan offers three theta function identities, two of which contain as special cases the celebrated two theorems of Ramanujan proved by Watson. Using addition formulas, the quintuple product identity, and a new general product formula for theta functions, we prove these three identities of Ramanujan from his lost notebooks.

     

Some extended results on a nonlinear ill-posed heat equation and remarks on a general case of nonlinear terms

The major object of this paper is to provide a quite convenient regularization method for a nonlinear backward heat problem. Error estimates for this method are provided together with a selection rule for the regularization parameter. Our method improve some results in a previous paper, including the earlier paper [D.D. Trong, N.H. Tuan, Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation, Nonlinear Anal. 71 (9) (2009) 4167–4176] and some other papers. A general case of nonlinear terms for this problem is obtained.     

Divisors of convergents of a continued fraction

By applying the theory of linear forms in logarithms, effective and sharpened versions of the results of Erdös and Mahler on the arithmetical properties of the convergents of a continued fraction are given.     

On a continued fraction of order 12

We present some new relations between a continued fraction U(q) of order 12 (established by M. S. M. Naika et al.) and U(q ⁿ ) for n = 7, 9, 11, 13:     

On a continued fraction of order six

In this paper, we derive certain identities for the following continued of order six:     

Explicit evaluations of a Ramanujan-Selberg continued fraction

This paper gives explicit evaluations for a Ramanujan-Selberg continued fraction in terms of class invariants and singular moduli.

     

Convergence and divergence of the Ramanujan AGM fraction

We prove that the Ramanujan AGM fraction diverges if |a|=|b| with a ²≠b ². Thereby we prove two conjectures posed by J. Borwein and R. Crandall. We also demonstrate a method for accelerating the convergence of this continued fraction when it converges.      

Some special results on convergent sequences of radon measures

Two problems will be considered. In Part I we consider a class of subsets of a topological space X and a Radon measure on X; if it is known that, for sufficiently many , the restrictions of the sets in constitutes a uniformity class in T w.r.t. the restriction of the given measure, then we ask if it follows that is a uniformity class in X.Part II, which can be read independently of Part I, is concerned with the question whether, to a given convergent sequence of Radon measures, say _n, there always exist sufficiently many compact sets K such that _n(K)(K).     

A multidimensional continued fraction based on a high-order recurrence relation

The paper describes and studies an iterative algorithm for finding small values of a set of linear forms over vectors of integers. The algorithm uses a linear recurrence relation to generate a vector sequence, the basic idea being to choose the integral coefficients in the recurrence relation in such a way that the linear forms take small values, subject to the requirement that the integers should not become too large. The problem of choosing good coefficients for the recurrence relation is thus related to the problem of finding a good approximation of a given vector by a vector in a certain one-parameter family of lattices; the novel feature of our approach is that practical formulae for the coefficients are obtained by considering the limit as the parameter tends to zero. The paper discusses two rounding procedures to solve the underlying inhomogeneous Diophantine approximation problem: the first, which we call ``naive rounding' leads to a multidimensional continued fraction algorithm with suboptimal asymptotic convergence properties; in particular, when it is applied to the familiar problem of simultaneous rational approximation, the algorithm reduces to the classical Jacobi-Perron algorithm. The second rounding procedure is Babai's nearest-plane procedure. We compare the two rounding procedures numerically; our experiments suggest that the multidimensional continued fraction corresponding to nearest-plane rounding converges at an optimal asymptotic rate.