Rogers-Ramanujan Continued Fraction Research Articles

General Formulas for Explicit Evaluations of Ramanujan's Cubic Continued Fraction

On page 366 of his lost notebook 15, Ramanujan recorded a cubic contin- ued fraction and several theorems analogous to Rogers-Ramanujan's continued fractions. In this paper, we derive several general formulas for explicit evaluations of Ramanujan's cubic continued fraction, several reciprocity theorems, two formulas connecting V (q) and V (q 3) and also establish some explicit evaluations using the values of remarkable product of theta-function.

ON THE INFINITE PRODUCTS DERIVED FROM THETA SERIES II

Let k be an imaginary quadratic field, <TEX>${\eta}$</TEX> the complex upper half plane, and let <TEX>${\tau}{\in}{\eta}{\cap}k,\;q=e^{{\pi}{i}{\tau}}$</TEX>. For n, t <TEX>${\in}{\mathbb{Z}}^+$</TEX> with <TEX>$1{\leq}t{\leq}n-1$</TEX>, set n=<TEX>${\delta}{\cdot}2^{\iota}$</TEX>(<TEX>${\delta}$</TEX>=2, 3, 5, 7, 9, 13, 15) with <TEX>${\iota}{\geq}0$</TEX> integer. Then we show that <TEX>$q{\frac}{n}{12}-{\frac}{t}{2}+{\frac}{t^2}{2n}{\prod}_{m=1}^{\infty}(1-q^{nm-t})(1-q^{{nm}-(n-t)})$</TEX> are algebraic numbers.

Erratum to "Modular curves and Ramanujan's continued fraction"

Erratum to "Modular curves and Ramanujan's continued fraction"

Ramanujan and extensions and contractions of continued fractions

If a continued fraction K =1 ∞ a n /b n is known to converge but its limit is not easy to determine, it may be easier to use an extension of K =1 ∞ a n /b n to find the limit. By an extension of K =1 ∞ a n /b n we mean a continued fraction K =1 ∞ c n /d n whose odd or even part is K =1 ∞ a n /b n . One can then possibly find the limit in one of three ways: We apply these ideas to derive new proofs of certain continued fraction identities of Ramanujan and to prove a generalization of an identity involving the Rogers-Ramanujan continued fraction, which was conjectured by Blecksmith and Brillhart.

Number theory in the spirit of Ramanujan

Introduction Congruences for $p(n)$ and $\tau(n)$ Sums of squares and sums of triangular numbers Eisenstein series The connection between hypergeometric functions and theta functions Applications of the primary theorem of Chapter 5 The Rogers-Ramanujan continued fraction Bibliography Index.

ARITHMETIC OF INFINITE PRODUCTS AND ROGERS-RAMANUJAN CONTINUED FRACTIONS

Let k be an imaginary quadratic field, h the complex upper half plane, and let <TEX>$\tau{\in}h{\cap}k$</TEX>, <TEX>$q=e^{{\pi}i\tau}$</TEX>. We find a lot of algebraic properties derived from theta functions, and by using this we explore some new algebraic numbers from Rogers-Ramanujan continued fractions.

The Rogers-Ramanujan continued fraction and a quintic iteration for $1/\pi$

Properties of the Rogers-Ramanujan continued fraction are used to obtain a formula for calculating $1/\pi$ with quintic convergence.

Bibasic $q$-Appell series connected with Ramanujan's continued fractions

In this paper we give some summation formulae for bibasic $ q $-Appell function $ \overline{\varphi}^{(1)} $ and express Appell function $ \overline{\varphi}^{(1)} $ in terms of classical continued fractions of Ramanujan. We also express the $ q $-Appell function $ \varphi^{(1)} $ on one base as a continued fraction. The bibasic approach for the Appell series, we think, is new.

ON THE INFINITE PRODUCTS DERIVED FROM THETA SERIES I

Let k be an imaginary quadratic field, h the complex upper half plane, and let <TEX>$\tau{\in}h{\cap}k,\;q=e^{{\pi}i\tau}$</TEX>. In this article, we obtain algebraic numbers from the 130 identities of Rogers-Ramanujan continued fractions investigated in [28] and [29] by using Berndt's idea ([3]). Using this, we get special transcendental numbers. For example, <TEX>$\frac{q^{1/8}}{1}+\frac{-q}{1+q}+\frac{-q^2}{1+q^2}+\cdots$</TEX> ([1]) is transcendental.

Ramanujan's cubic continued fraction revisited

In this article, we derive a sequence of numbers which converge to 1/π. We will also derive a new series for 1/π. These new results are motivated by the study of Ramanujan’s cubic continued fraction.

A q-CONTINUED FRACTION

We use the method of generating functions to find the limit of a q-continued fraction, with 4 parameters, as a ratio of certain q-series. We then use this result to give new proofs of several known continued fraction identities, including Ramanujan's continued fraction expansions for (q2; q3)∞/(q; q3)∞and [Formula: see text]. In addition, we give a new proof of the famous Rogers–Ramanujan identities. We also use our main result to derive two generalizations of another continued fraction due to Ramanujan.

Errata: Ramanujan's Continued Fraction for a Puzzle

Errata: Ramanujan's Continued Fraction for a Puzzle

The convergence behavior of q-continued fractions on the unit circle

In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of $q$-continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each $q$-continued fraction, $G(q)$, in this class, that there is an uncountable set of points, $Y_{G}$, on the unit circle such that if $y \in Y_{G}$ then $G(y)$ does not converge to a finite value. We discuss the implications of our theorems for the convergence of other $q$-continued fractions, for example the G\"ollnitz-Gordon continued fraction, on the unit circle.

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 = e2π 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.

Modular curves and Ramanujan's continued fraction

Modular curves and Ramanujan's continued fraction

Ramanujan's Continued Fraction for a Puzzle

Ramanujan's Continued Fraction for a Puzzle

The explicit formulas and evaluations of Ramanujan's theta-function ψ

We define two quotients of theta-function ψ depending on two positive real parameters. We then show how they are connected with two parameters of Dedekind eta-function, theta-function φ, and the Ramanujan–Weber class invariants. Explicit formulas for determining values of the theta-function ψ are derived, and several examples will be given. In addition, we give some applications of these parameters for the famous Rogers–Ramanujan continued fraction R ( q ) , Ramanujan's cubic continued fraction G ( q ) , and the modular j-invariant.

Some Continued Fractions in Ramanujan’s Lost Notebook

In this paper we first give the value of a periodic continued fraction which was recorded incorrectly by Ramanujan on page 341 of his lost notebook. Next, we describe several pairs of equivalent continued fractions in which one is the odd part of the other. One of the results is for the Rogers-Ramanujan continued fraction which was recently proved by Berndt and Yee. Finally, using the Bauer-Muir transformation we prove the equivalence of two continued fractions. One was recorded on page 44 in Ramanujan’s lost notebook, and the other is found in the unorganized pages at the end of Ramanujan’s second notebook.

A new approach to Jacobi's theorems via Ramanujan's continued fractions

In this paper, we show that Jacobi's two-square and two-triangular number theorems are immediate consequence of Ramanujan's continued fraction identities.

Continued fractions and modular functions

It is widely recognized that the work of Ramanujan deeply influenced the direction of modern number theory. This influence resonates clearly in the “Ramanujan conjectures.” Here I will explore another part of his work whose position within number theory seems to be less well understood, even though it is more elementary, namely that related to continued fractions. I will concentrate on the special values of continued fractions that represent modular functions, especially the Rogers-Ramanujan continued fraction. These give analogues of the simple continued fraction expansions of units in real quadratic fields. My primary motivation is to furnish a coherent treatment of this topic, around which an air of mystery seems to linger. Another is to provide an inviting and non-standard introduction to the classical theory of modular functions. This is largely an expository paper; most of the ideas I discuss are well known. Yet it is hoped that the elaboration given here combines these ideas in a novel way. Although this paper is not intended to be comprehensive, its later sections contain more material than is likely needed to gain a clear impression of the main themes, which the first six sections should provide. These will take the general reader through a proof of the first main result, Theorem 1, introducing the needed concepts along the way. The sections that follow these assume a somewhat greater background in number theory.