相关论文: Notes On a Continued Fraction of Ramanujan

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.

We revisit an infinitely nested radical by Ramanujan. Utilizing the full strength of his method, we shall arrive at some new infinitely nested radicals.

We present a general introduction to continued fractions, with special consideration to the function fields case. These notes were prepared for a summer class given this year in Beijing at Beihang university.

This note is a complement to an article which was published, six years ago, in The Ramanujan Journal (vol. 45.3, 2018). Here, the goal is to fully describe a singular transcendental continued fraction in Q((T^-1)), tied to a particular…

A continued fraction $v(\tau)$ of Ramanujan is evaluated at certain arguments in the field $K = \mathbb{Q}(\sqrt{-d})$, with $-d \equiv 1$ (mod $8$), in which the ideal $(2) = \wp_2 \wp_2'$ is a product of two prime ideals. These values of…

In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions.

We make a summary of the different types of proofs adding some new ideas. In addition we conjecture some relations which could be necessary in "modular type proofs" (not still found) of the Ramanujan-like series for 1/\pi^2.

In this paper, we consider a general form of the analogue of Ramanujan's sum in the ring of polynomials over a finite field. We first prove some multiplicative properties of such functions before considering their finite Fourier series and…

We highlight some facts about continued fractions of real cubic irrationalities. This may be thought as a small section in a textbook on continued fractions.

Ramanujan recorded four reciprocity formulas for the Roger-Ramanujan continued fraction. Two reciprocity formulas each are also associated with the Ramanujan--G\"ollnitz--Gordon continued fraction and a level-13 analog of the…

In this article we derive an approximation inequality for continued radicals, generalizing an inequality of Herschfeld for continued square roots to arbitrary radicals, which is useful in exploring convergence issues and obtaining…

The Ramanujan Machine project detects new expressions related to constants of interest, such as $\zeta$ function values, $\gamma$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures concerning the…

We suggest a continued fraction origin to Ramanujan's approximation to {(a-b)/(a+b)}^2 in terms of the arc length of an ellipse with semiaxes a and b. Moreover, we discuss the asymptotic accuracy of the approximation.

Abstrct: In this note, by considering fractionally linear functions over a finite field and consequently developing an abstract sequence, we study some of its properties.

By using Euler's approach of using Euclid's algorithm to expand a power series into a continued fraction, we show how to derive Ramanujan's $q$-continued fractions in a systematic manner.

The authors provide a survey of certain aspects of their joint work with the late M. K. Vamanamurthy. Most of the results are simple to state and deal with special functions, a topic of research where S. Ramanujan's contributions are…

In this work, we present continued fractions for the arithmetic, geometric, harmonic and cotangent means of $[a_0,a_1,\dots,a_k]$ and $[a_0,a_1,\dots,a_k,a_{k+1}]$, and some of their applications.

This paper is a short survey of the recent results on examples of periodic two-dimensional continued fractions (in Klein's model). In the last part of this paper we formulate some questions, problems and conjectures on geometrical…

We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical…

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…