Modular forms arising from divisor functions

For an infinite family of modular forms constructed from Klein forms we provide certain explicit formulas for their Fourier coefficients by using the theory of basic hypergeometric series (Theorem 2). By making use of these modular forms we investigate the bases of the vector spaces of modular forms of some levels (Theorem 5) and find its application.     

Hypergeometric series and continued fractions

Ramanujan’s results on continued fractions are simple consequences of three-term relations between hypergeometric series. Theirq-analogues lead to many of the continued fractions given in the ‘Lost’ notebook in particular the famous one considered by Andrews and others.     

Continued fractions with three limit points

On page 45 in his lost notebook, Ramanujan asserts that a certain q-continued fraction has three limit points. More precisely, if A_n/B_n denotes its nth partial quotient, and n tends to ∞ in each of three residue classes modulo 3, then each of the three limits of A_n/B_n exists and is explicitly given by Ramanujan. Ramanujan's assertion is proved in this paper. Moreover, general classes of continued fractions with three limit points are established.     

Identities of the Rogers-Ramanujan-Bailey type

A multiparameter generalization of the Bailey pair is defined in such a way as to include as special cases all Bailey pairs considered by W.N. Bailey in his paper [Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 50 (1949) 421-435]. This leads to the derivation of a number of elegant new Rogers-Ramanujan type identities.     

On some generalizations of Ramanujan’s continued fraction identities

In this note we establish continued fraction developments for the ratios of the basic hypergeometric function₂ϕ₁(a,b;c;x) with several of its contiguous functions. We thus generalize and give a unified approach to establishing several continued fraction identities including those of Srinivasa Ramanujan.     

Modular forms which behave like theta series

In this paper, we determine all modular forms of weights , , for the full modular group which behave like theta series, i.e., which have in their Fourier expansions, the constant term and all other Fourier coefficients are non-negative rational integers. In fact, we give convex regions in (resp. in ) for the cases (resp. for the cases ). Corresponding to each lattice point in these regions, we get a modular form with the above property. As an application, we determine the possible exceptions of quadratic forms in the respective dimensions.

     

Simple continued fractions and cutting sequences

Abstract

The connection between cutting sequences of a directed geodesic in the tessellated hyperbolic plane ?², the modular group Γ = PSL(2, ?) and the simple continued fractions of an end point w of the geodesic have been established by Series [13]. In this paper we represent the simple continued fractions of w ∈ ? and the “L” and “R” codes of the cutting sequence in terms of modular and extended modular transformations. We will define a T ₀-path on a graph whose vertices are the set of Farey triangles, as the equivalent of the cutting sequence. The relationship between the directed geodesic with end point w on ?, the Farey tessellation and the simple continued fraction expansion of w ∈ ? _∞ then follows easily as a consequence of this redefinition. Finite, infinite and periodic simple continued fractions are subsequently examined in this light.     

Partitions: At the Interface of q-Series and Modular Forms

In this paper we explore five topics from the theory of partitions: (1) the Rademacher conjecture, (2) the Herschel-Cayley-Sylvester formulas, (3) the asymptotic expansions of E.M. Wright, (4) the asymptotics of mock theta function coefficients, (5) modular transformations of q-series.     

Explicit continued fractions with expected partial quotient growth

For let be the continued fraction expansion of . Write

We construct some numbers 's with

     

q-Difference equation and the Cauchy operator identities

In this paper, we verify the Cauchy operator identities by a new method. And by using the Cauchy operator identities, we obtain a generating function for Rogers-Szegö polynomials. Applying the technique of parameter augmentation to two multiple generalizations of q-Chu-Vandermonde summation theorem given by Milne, we also obtain two multiple generalizations of the Kalnins-Miller transformation.     

An algorithm of infinite sums representations and Tasoev continued fractions

For any given real number, its corresponding continued fraction is unique. However, given an arbitrary continued fraction, there has been no general way to identify its corresponding real number. In this paper we shall show a general algorithm from continued fractions to real numbers via infinite sums representations. Using this algorithm, we obtain some new Tasoev continued fractions.

     

Two operator identities and their applications to terminating basic hypergeometric series and q-integrals

In this paper, we first give two interesting operator identities, and then, using them and the q-exponential operator technique to some terminating summation formulas of basic hypergeometric series and q-integrals, we obtain some q-series identities and q-integrals involving ₃?₂.     

Rates of convergence for continued fractions of irrational numbers

The dynamics of the Gauss Map suggests a way to compare the convergence to a real number ζ ε(0,l) of a continued fraction and the divergence of the orbit of ζ Of particular interest is the comparison of the rate of convergence to ζ of its simple continued fraction and the rate of divergence by the Gauss Map of the orbit of ζ for all irrational numbers in (0,l). We state and prove sharp inequalities for the convergence of the sequence of rational convergents of an irrational number ζ. We show that the product of the rate of convergence of the continued fraction of ζ and the rate of divergence by the Gauss Map of the orbit of ζ equals 1.     

On the values of continued fractions: q-series

Using the Poincaré-Perron theorem on the asymptotics of the solutions of linear recurrences it is proved that for a class of q-continued fractions the value of the continued fraction is given by a quotient of the solution and its q-shifted value of the corresponding q-functional equation.     

Modular forms and differential operators

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]_n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]_n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ² + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree). Dedicated to the memory of Professor K G Ramanathan     

A note on bounds for the derivatives of continued fractions

Although it is difficult to differentiate analytic functions defined by continued fractions, it is relatively easy in some cases to determine uniform bounds on such derivatives by perceiving the continued fraction as an infinite composition of linear fractional transformations and applying an infinite chain rule for differentiation.     

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.     

On suborbital graphs and related continued fractions

In this paper, we study suborbital graphs for congruence subgroup Γ₀(n) of the modular group Γ to have hyperbolic paths of minimal lengths. It turns out that these graphs give rise to a special continued fraction which is a special case of very famous fraction coming out from Pringsheim’s theorem.     

q-Differential operator identities and applications

In this paper, we construct a new q-exponential operator and obtain some operator identities. Using these operator identities, we give a formal extension of Jackson's transformation formula. A formal extension of Bailey's summation and an extension of the Sears terminating balanced transformation formula are also derived by our operator method. In addition, we also derive several interesting a formal extensions involving multiple sum about three terms of Sears transformation formula and Heine's transformation formula.