Rogers-Ramanujan Continued Fraction Research Articles

Modularity of a certain continued fraction of Ramanujan

Modularity of a certain continued fraction of Ramanujan

Distribution and divisibility of the Fourier coefficients of certain Hauptmoduln

Suppose $j_N(\tau)$ and $j_N^{*}(\tau)$ are the Hauptmoduln of the congruence subgroup $\Gamma_0(N)$ and the Fricke group $\Gamma^{*}_0(N)$, respectively. In [7], the authors predicted that, like Klein's $j$-function, the Fourier coefficients of $j_N(\tau)$ and $j_{N}^{*}(\tau)$ in some arithmetic progression are both even and odd with density $\frac{1}{2}$. In this article, we can find some arithmetic progression of $n$ where the Fourier coefficients of $j_6(\tau)$ (resp. $j_6^{*}(\tau)$ and $j_{10}(\tau)$) are almost always even. Furthermore, using Hecke eigenforms and Rogers-Ramanujan continued fraction, we obtain infinite families of congruences for $j_6(\tau)$, $j_6^{*}(\tau)$, $j_{10}(\tau),$ and $j_{10}^{*}(\tau)$.

On Generalized Mock Theta Functions of Tenth Order

In this paper we have given 3 independent variable generalization of tenth order mock theta functions and have found some relations between generalized mock theta functions of different order. We have given a continued fraction representation of generalized tenth order mock theta functions. We have also given the q-integral representation and multibasic expansion for these functions. Further by specializing the parameters we have connected mock theta functions with continued fraction of Ramanujan.
 HIGHLIGHTS
 
 Mock theta functions are last gift to mathematics by Ramanujan. These functions are also called mysterious functions as the exact meaning of order of mock theta functions is mystery for mathematicians
 The generalization of tenth order mock theta functions connects them with different order mock theta functions and with Ramanujan’s continued fractions
 The specialization of parameters of generalized mock theta functions gives the famous Ramanujan’s continued fraction

On the coefficients of a continued fraction of Ramanujan

On expanding Ramanujan's continued fraction in power series, we observe that the sign of the coefficients is periodic with period 4. We also give a combinatorial interpretation for the coefficients.

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

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

English

In 1998, Michael Hirschhorn discovered the 5-dissection formulas of the Rogers-Ramanujan continued fraction R(q) and its reciprocal. We obtain the 5-dissections for functions R(q)R(q2)2 and R(q)2/R(q2), which are essentially Ramanujan’s parameter and its companion. Additionally, 5-dissections of the reciprocals of these two functions are derived. These 5-dissection formulas imply that the coefficients in their series expansions have periodic sign patterns with few exceptions.

Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

On Ramanujan’s Incomplete Elliptic Integral Identities

In his lost notebook, S. Ramanujan recorded incomplete elliptic integral identities of the first kind. In this paper, we give new proofs to Ramanujan’s incomplete elliptic integrals of the first kind for level 5 using the parameter k(q) = R(q)R2(q2), where R(q) is Rogers-Ramanujan continued fraction. Also, we construct new identities for level 5 similar to ones found in Ramanujan’s work. Further, we prove related identities of level 7, using theta function identities.

New theta function identities for a continued fraction of Ramanujan and their applications

We prove three new theta function identities for the continued fraction H(q) defined by $$\begin{aligned} H(q):=q^{1/8}-\frac{q^{7/8}}{1-q}_{+}\frac{q^2}{1+q^2}_{-}\frac{q^3}{1-q^3}_{+}\frac{q^4}{1+q^4}_{- \cdots }, \vert q\vert <1. \end{aligned}$$ The theta-function identities are then used to prove integral representations for the continued fraction H(q). We also prove general theorems and reciprocity formulas for the explicit evaluation of the continued fraction H(q). The results are analogous to those of the famous Rogers-Ramanujan continued fraction.

On Entry II.16.12: A continued fraction of Ramanujan

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 Ramanujan’s Entry 12, one using a method of Euler, and another using the theory of orthogonal polynomials. We consider a natural generalization of Entry 12 suggested by the theory of orthogonal polynomials.

Evaluations of the cubic continued fraction by some theta function identities

In this paper, we use some theta function identities involving two parameters $h_{n,k}$ and $h'_{n,k}$ for the theta function $\varphi$ to establish new evaluations of Ramanujan's cubic continued fraction.

Infinite families of congruences modulo 5 and 7 for the cubic partition function

AbstractIn 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.

Relations Between Ramanujan's Cubic Continued Fraction and a Continued Fraction of Order 12 and its Evaluations

Relations Between Ramanujan's Cubic Continued Fraction and a Continued Fraction of Order 12 and its Evaluations

Some results on a special case of a general continued fraction of Ramanujan

We derive a new special case C(q) of a general continued fraction recorded by Ramanujan in his Lost Notebook. We give a representation of the continued fraction C(q) as a quotient of Dedekind eta-function and then use it to prove modular identities connecting C(q) with each of the continued fractions $$C(-q)$$ , $$C(q^{2})$$ , $$C(q^{3})$$ , $$C(q^{5})$$ , $$C(q^{7})$$ , $$C(q^{11})$$ , $$C(q^{13})$$ and $$C(q^{17})$$ . We also prove general theorems for the explicit evaluation of the continued fraction C(q) by using Ramanujan’s class invariants.

A continued fraction of Ramanujan and some Ramanujan-Weber class invariants

On Page 36 of his ``lost notebook, Ramanujan recorded four $q$-series representations of the famous Rogers-Ramanujan continued fraction. In this paper, we establish two $q$-series representations of Ramanujan's continued fraction found in his ``lost notebook. We also establish three equivalent integral representations and modular equations for a special case of this continued fraction. Furthermore, we derive continued-fraction representations for the Ramanujan-Weber class invariants $g_n$ and $G_n$ and establish formulas connecting $g_n$ and $G_n$. We obtain relations between our continued fraction with the Ramanujan-G\{o}llnitz-Gordon and Ramanujan's cubic continued fractions. Finally, we find some algebraic numbers and transcendental numbers associated with a certain continued fraction $A(q)$ which is related to Ramanujan's continued fraction $F(a,b,\lambda;q),$ the Ramanujan-G\{o}llnitz-Gordon continued fraction $H(q)$ and the Dedekind eta function $\eta(s)$.

New theta-function identities and general theorems for the explicit evaluations of Ramanujan’s continued fractions

We prove some new theta-function identities for two continued fractions of Ramanujan which are analogous to those of Ramanujan–Gollnitz–Gordon continued fraction. Then these identities are used to prove new general theorems for the explicit evaluations of the continued fractions.

A framework of Rogers–Ramanujan identities and their arithmetic properties

The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+\sigma)}}{(1-q)\cdots (1-q^n)}, \] where $\sigma=0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers-Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers-Ramanujan identities to doubly-infinite families of $q$-series identities. If $a\in\{1,2\}$ and $m,n\geq 1$, then we have \[ \sum_{\substack{\lambda \lambda_1\leq m}} q^{a|\lambda|} P_{2\lambda}(1,q,q^2,\dots;q^n) =\textrm{"infinite product modular function"}, \] where the $P_{\lambda}(x_1,x_2,\dots;q)$ are Hall-Littlewood polynomials. These $q$-series are specialized characters of affine Kac--Moody algebras. Generalizing the Rogers-Ramanujan continued fraction, we prove in the case of $\textrm{A}_{2n}^{(2)}$ that the relevant $q$-series quotients are integral units.

Generalized reciprocal identities

Included in Ramanujan’s Notebooks are two reciprocal identities. The first identity connects the Rogers-Ramanujan continued fraction with an eta quotient. The second identity is a level thirteen analogue. These are special cases of a more general class of relations between eta quotients and modular functions defined by product generalizations of the Rogers-Ramanujan continued fraction. Each identity is shown to be a relation between generators for a certain congruence subgroup. The degree, form, and symmetry of the identities is determined from behavior at cusps of the congruence subgroup whose field of functions the parameters generate. The reciprocal identities encode information about fundamental units and class numbers for real quadratic fields.

Convergence properties of the classical and generalized Rogers-Ramanujan continued fraction

The aim of this paper is to study the convergence and divergence of the Rogers-Ramanujan and the generalized Rogers-Ramanujan continued fractions on the unit circle. We provide an example of an uncountable set of measure zero on which the Rogers-Ramanujan continued fraction R(x) diverges and which enlarges a set previously found by Bowman and Mc Laughlin. We further study the generalized Rogers-Ramanujan continued fractions R a (x) for roots of unity a and give explicit convergence and divergence conditions. As such, we extend some work of Huang towards a question originally investigated by Ramanujan and some work of Schur on the convergence of R(x) at roots of unity. In the end, we state several conjectures and possible directions for generalizing Schur’s result to all Rogers-Ramanujan continued fractions R a (x). 2010 Mathematics Subject Classification 11A55, 11P84

New General Theorems and Explicit Values of the Level 13 Analogue of Rogers-Ramanujan Continued Fraction

We prove general theorems for the explicit evaluations of the level 13 analogue of Rogers-Ramanujan continued fraction and find some new explicit values. This work is a sequel to some recent works of S. Cooper and D. Ye.