On Ramanujan's continued fraction

Domains of validity for some of Ramanujan's continued fraction formulas

Ramanujan, who loved continued fractions, recorded many of his formulas in his two notebooks. The 25th entry in Chapter 12 of Notebook II is a continued fraction involving a quotient of gamma function products. We are going to give a new proof of Entry 25, which, incidently, is the least difficult of all Ramanujan's continued fraction formulas involving quotients of gamma function products. In our proof, we make use of a hypergeometric formula of his.

Ramanujan recorded many beautiful continued fractions in his notebooks. In this paper, we derive several indentities involving the Ramanujan continued fraction c(q), including relations between c(q) and c(qn). We also obtain explicit evaluations of for various positive integers n.

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

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.

In 2010, Hei-Chi Chan introduced the cubic partition function a(n) in connection with Ramanujan's cubic continued fraction. Chen and Lin, and Ahmed, Baruah and Dastidar proved that a(25n + 22) ≡ 0 (mod 5) for n ⩾ 0. In this paper, we prove several infinite families of congruences modulo 5 and 7 for a(n). Our results generalize the congruence a(25n + 22) ≡ 0 (mod 5) and four congruences modulo 7 for a(n) due to Chen and Lin. Moreover, we present some non-standard congruences modulo 5 for a(n) by using an identity of Newman. For example, we prove that $a((({15\times 17^{3\alpha }+1})/{8})) \equiv 3^{\alpha +1} \ ({\rm mod}\ 5)$ for α ⩾ 0.

On a continued fraction of Ramanujan, two and four-square theorems and bibasic q-Appell functions

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In 1961, Baker, Gammel and Wills conjectured that for functions f meromorphic in the unit ball, a subsequence of its diagonal Pade approximants converges uniformly in compact subsets of the ball omitting poles of f. There is also apparently a cruder version of the conjecture due to Pade himself, going back to the early twentieth century. We show here that for carefully chosen q on the unit circle, the Rogers-Ramanujan continued fraction

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

Set , where is an irrational number, and let be the radius of holomorphy of the Rogers-Ramanujan function As is known, and for each there exists such that . It is proved here that the function is meromorphic not only in the disc , but also in the disc , which is larger for ; and that the Rogers-Ramanujan continued fraction converges to on compact subsets contained in , where is the union of circles with centres at and passing through the poles of . The convergence of the Rogers-Ramanujan continued fraction in the domain was established earlier by Lubinsky.

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.

On a sequence of continued fractions of Ramanujan and theirq-analogs

In this paper, we present the natural extension of the Rogers-Ramanujan continued fraction to the nonterminating very well-poised basic hypergeometric function 8 ϕ 7 _8{\phi _7} . In a letter to Hardy, Ramanujan indicated that he possessed a four variable generalization. Our generalization has seven variables and is, perhaps, all one can expect from this method.

In his first and second letters to Hardy, Ramanujan made several assertions about the Rogers-Ramanujan continued fraction F ( q ) F(q) . In order to prove some of these claims, G. N. Watson established two important theorems about F ( q ) F(q) 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.