Related papers: Ramanujan and the Regular Continued Fraction Expan…

We prove some new modular identities for the Rogers\textendash Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers\textendash Ramanujan continued fraction, then…

We consider continued fractions \frac{-a_1}{1-\frac{a_2}{1-\frac{a_3}{1-...}}} \label{fr} with real coefficients $a_i$ converging to a limit $a$. S.Ramanujan had stated the theorem (see [ABJL], p.38) saying that if $a\neq\frac14$, then the…

We extend a result of Han\v{c}l, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence $\{\alpha_n\}$ of algebraic integers of bounded degree, each attaining the maximum absolute value…

We study Ramanujan's cubic continued fraction and explicit evaluations of theta-functions

We consider series of the form $$ \frac{p}{q} +\sum_{j=2}^\infty \frac{1}{x_j}, $$ where $x_1=q$ and the integer sequence $(x_n)$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n|x_{n+1}$ for $n\geq 1$.…

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:…

Adler, Keane, and Smorodinsky showed that if one concatenates the finite continued fraction expansions of the sequence of rationals \[ \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{1}{5}, \cdots \] into…

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 the present work, we established continued fractions of level eighteen, twenty six and thirty. Further, we obtained vanishing coefficients and many algebraic relations. To validate our result colored partitions are also obtained.

In this article we present evaluations of continued fractions studied by Ramanujan. More precisely we give the complete polynomial equations of Rogers-Ramanujan and other continued fractions, using tools from the elementary theory of the…

We observe that certain famous evaluations of the Rogers-Ramanujan continued fraction $R(q)$ are close to $2\pi-6$ and $(2\pi-6)/2\pi$, and that $2\pi-6$ can be expressed by a Rogers-Ramanujan continued fraction in which $q$ is very nearly…

It is widely believed that the continued fraction expansion of every irrational algebraic number $\alpha$ either is eventually periodic (and we know that this is the case if and only if $\alpha$ is a quadratic irrational), or it contains…

The primary purpose of this paper is to provide a survey of properties, values, identities, and generalizations of the Rogers--Ramanujan continued fraction, which is closely related to the Rogers--Ramanujan identities. Many of these results…

We investigate integer numbers which possess at the same time the properties to be triangulars and squares, that are, numbers $a$ for which do exist integers $m$ and $n$ such that $ a = n^2 = \frac{m \cdot (m+1)}{2} $. In particular, we are…

Let $\omega=[a_1, a_2, \cdots]$ be the infinite expansion of continued fraction for an irrational number $\omega \in (0,1)$; let $R_n (\omega)$ (resp. $R_{n, \, k} (\omega)$, $R_{n, \, k+} (\omega)$) be the number of distinct partial…

Let $R(q)$ be the Rogers-Ramanujan continued fraction. We give different proofs of two complementary relations for $R(q)$ given by Ramanujan and proved by Watson and Ramanathan. Our proofs only use product expansions for classical Jacobi…

In this paper, we clarified the relationship between continued fractions, determinants, and identities, making it easier to apply these methods systematically in other settings. In particular, we studied finite continued fractions from the…

In this paper, we will first summarize known results concerning continued fractions. Then we will limit our consideration to continued fractions of quadratic numbers. The second author described periods and sometimes precise form of…

We study a continued fraction due to Ramanujan, that he recorded as Entry 12 in Chapter 16 of his second notebook. It is presented in Part III of Berndt's volumes on Ramanujan's notebooks. We give two alternate approaches to proving…

We deduce $q$-continued fractions $S_{1}(q)$, $S_{2}(q)$ and $S_{3}(q)$ of order fourteen, and continued fractions $V_{1}(q)$, $V_{2}(q)$ and $V_{3}(q)$ of order twenty-eight from a general continued fraction identity of Ramanujan. We…