On quintic Eisenstein series and points of order five of the Weierstrass elliptic functions

We employ a new constructive approach to study modular forms of level five by evaluating the Weierstrass elliptic functions at points of order five on the period parallelogram. A significant tool in our analysis is a nonlinear system of coupled differential equations analogous to Ramanujan??s differential system for the Eisenstein series on SL(2,?). The resulting relations of level five may be written as a coupled system of differential equations for quintic Eisenstein series. Some interesting combinatorial and analytic consequences result, including an alternative proof of a famous identity of Ramanujan involving the Rogers?CRamanujan continued fraction.     

Eisenstein series and theta functions to the septic base

We develop a theory for Eisenstein series to the septic base, which was started by S. Ramanujan in his “Lost Notebook.” We show that two types of septic Eisenstein series may be parameterized in terms of the septic theta function and the eta quotient η⁴(7τ)/η⁴(τ). This is accomplished by constructing elliptic functions which have the septic Eisenstein series as Taylor coefficients. The elliptic functions are shown to be solutions of a differential equation, and this leads to a recurrence relation for the septic Eisenstein series.     

Differential equations for septic theta functions

We demonstrate that quotients of septic theta functions appearing in Ramanujan’s Notebooks and in Klein’s work satisfy a new coupled system of nonlinear differential equations with symmetric form. This differential system bears a close resemblance to an analogous system for quintic theta functions. The proof extends an elementary technique used by Ramanujan to prove the classical differential system for normalized Eisenstein series on the full modular group. In the course of our work, we show that Klein’s quartic relation induces symmetric representations for low-weight Eisenstein series in terms of weight one modular forms of level seven.     

Non-linear algebraic differential equations satisfied by a certain family of elliptic functions

Kurokawa and Wakayama (Ramanujan J. 10:23–41, 2005) studied a family of elliptic functions defined by certain q-series. This family, in particular, contains the Weierstrass ?-function. In this paper, we prove that elliptic functions in this family satisfy certain non-linear algebraic differential equations whose coefficients are essentially given by rational functions of the first few Eisenstein series of the modular group.     

On multiple series of Eisenstein type

The aim of this paper is to study certain multiple series which can be regarded as multiple analogues of Eisenstein series. As part of a prior research, the second-named author considered double analogues of Eisenstein series and expressed them as polynomials in terms of ordinary Eisenstein series. This fact was derived from the analytic observation of infinite series involving hyperbolic functions which were based on the study of Cauchy, and also Ramanujan. In this paper, we prove an explicit relation formula among these series. This gives an alternative proof of this fact by using the technique of partial fraction decompositions of multiple series which was introduced by Gangl, Kaneko and Zagier. By the same method, we further show a certain multiple analogue of this fact and give some examples of explicit formulas. Finally we give several remarks about the relation between the results of the present and the previous works for infinite series involving hyperbolic functions.     

Hadamard products for generalized Rogers–Ramanujan series

The purpose of this paper is to derive product representations for generalizations of the Rogers–Ramanujan series. Special cases of the results presented here were first stated by Ramanujan in the “Lost Notebook” and proved by George Andrews. The analysis used in this paper is based upon the work of Andrews and the broad contributions made by Mourad Ismail and Walter Hayman. Each series considered is related to an extension of the Rogers–Ramanujan continued fraction and corresponds to an orthogonal polynomial sequence generalizing classical orthogonal sequences. Using Ramanujan's differential equations for Eisenstein series and corresponding analogues derived by V. Ramamani, the coefficients in the series representations of each zero are expressed in terms of certain Eisenstein series.     

New indefinite integrals of Heun functions

We present a conspicuous number of indefinite integrals involving Heun functions and their products obtained by means of the Lagrangian formulation of a general homogeneous linear ordinary differential equation. As a by-product we also derive new indefinite integrals involving the Gauss hypergeometric function and products of hypergeometric functions with elliptic functions of the first kind. All integrals we obtained cannot be computed using Maple and Mathematica.     

On differential modular forms and some analytic relations between Eisenstein series

In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight 2, 4 and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein series and obtain them in a natural way as coefficients of a family of elliptic curves. The fact that a complex manifold over the moduli of polarized Hodge structures in the case h ¹⁰=h ⁰¹=1 has an algebraic structure with an action of an algebraic group plays a basic role in all of the proofs.      

Analogues of the Hurwitz Formulas for Level 2 Eisenstein Series

In this paper, we consider certain double series of Eisenstein type involving hyperbolic functions, which can be regarded as analogues of the level 2 Eisenstein series. We prove some evaluation formulas for these series at positive integers which are analogues of both the Hurwitz formulas for the level 2 Eisenstein series and the classical results given by Cauchy, Lerch, Mellin and Ramanujan.     

Integrals of <Emphasis Type="Italic">K</Emphasis> and <Emphasis Type="Italic">E</Emphasis> from lattice sums

We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals K and E. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.     

On Chudnovsky–Ramanujan type formulae

In a well known 1914 paper, Ramanujan gave a number of rapidly converging series for \(1/\pi \) which are derived using modular functions of higher level. Chudnovsky and Chudnovsky in their 1988 paper derived an analogous series representing \(1/\pi \) using the modular function J of level 1, which results in highly convergent series for \(1/\pi \), often used in practice. In this paper, we explain the Chudnovsky method in the context of elliptic curves, modular curves, and the Picard–Fuchs differential equation. In doing so, we also generalize their method to produce formulae which are valid around any singular point of the Picard–Fuchs differential equation. Applying the method to the family of elliptic curves parameterized by the absolute Klein invariant J of level 1, we determine all Chudnovsky–Ramanujan type formulae which are valid around one of the three singular points: \(0, 1, \infty \).     

Basic representations for Eisenstein series from their differential equations

In this paper we provide a new approach for the derivation of parameterizations for the Eisenstein series. We demonstrate that a variety of classical formulas may be derived in an elementary way, without knowledge of the inversion formulae for the corresponding Schwarzian triangle functions. In particular, we provide a new derivation for the parametric representations of the Eisenstein series in terms of the complete elliptic integral of the first kind. The proof given here is distinguished from existing elementary proofs in that we do not employ the Jacobi-Ramanujan inversion formula relating theta functions and hypergeometric series. Our alternative approach is based on a Lie symmetry group for the differential equations satisfied by certain Eisenstein series. We employ similar arguments to obtain parameterizations from Ramanujan's alternative signatures and those associated with the inversion formula for the modular J-function. Moreover, we show that these parameterizations represent the only possible signatures under a certain assumed form for the Lie group parameters.     

On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type

In this paper, we consider certain analogues of Eisenstein seriesinvolving hyperbolic sine and cosine functions. Using a resultof Hurwitz, we prove some evaluation formulas for these seriesat positive integers. We can regard these formulas as analoguesof the famous formulas for certain series given by Cauchy, Ramanujan,Berndt, and so on, as well as those for the Eisenstein seriesgiven by Hurwitz.     

The elliptic Apostol–Dedekind sums generate odd Dedekind symbols with Laurent polynomial reciprocity laws

Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol–Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent polynomial reciprocity laws. Our construction is based on Machide’s result (J Number Theory 128:1060–1073, 2008) on his elliptic Dedekind–Rademacher sums. As an application of our results, we discover Eisenstein series identities which generalize certain formulas by Ramanujan (Collected Papers of Srinivasa Ramanujan, pp. 136–162. AMS Chelsea Publishing, Providence, 2000), van der Pol (Indag Math 13:261–271, 272–284, 1951), Rankin (Proc R Soc Edinburgh Sect A 76:107–117, 1976) and Skoruppa (J Number Theory 43:68–73, 1993).     

Srinivasa Ramanujan (1887‐1920) A centennial tribute

This centennial tribute commemorates Ramanujan the man and Ramanujan the mathematician. A brief account of his remarkable mathematical contributions is presented to describe the great talent of Srinivasa Ramanujan. Considerable discussion is made of his new and important ideas, results and theorems in various subjects including definite integrals, hypergeometric series, Bernoulli and Euler numbers, the Riemann zeta function, distribution of prime numbers, theory of partitions and statistical mechanics, continued fractions, hypergeometric functions, Rogers‐Ramanujan identities, theory of representation of numbers as sums of squares, Ramanujan's τ‐function, q‐series, elliptic and modular functions.     

First integrals of difference equations which do not possess a variational formulation

The paper presents a new method for finding first integrals of difference equations which do not possess Lagrangians, nor Hamiltonians. In this paper we consider ordinary differential and difference equations. As an example we solve a third order nonlinear ordinary differential equation and its invariant discretization using three first integrals obtained by means of this method.     

Eisenstein series and convolution sums

The Ramanujan Journal - We compute Fourier series expansions of weight 2 and weight 4 Eisenstein series at various cusps. Then we use results of these computations to give formulas for the...     

Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions

Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the independent variable u is the argument of Jacobian elliptic function theory. This allows a limitless number of indefinite integrals with respect to the amplitude to be derived for these three elliptic integrals. Sample results are given, including the integrals derived from powers of the 12 Glaisher elliptic functions. New recurrence relations and integrals are also given for the 12 Glaisher elliptic functions.     

Asymptotic expansion of Fourier coefficients of reciprocals of Eisenstein series

The Ramanujan Journal - In this paper we give a classification of the asymptotic expansion of the q-expansion of reciprocals of Eisenstein series $$E_k$$ of weight k for the modular group $$\mathop...     

On nonlinear systems consisting of different types of differential equations

We consider a system consisting of a quasilinear parabolic equation and a first order ordinary differential equation where both equations contain functional dependence on the unknown functions. Then we consider a system which consists of a quasilinear parabolic partial differential equation, a first order ordinary differential equation and an elliptic partial differential equation. These systems were motivated by models describing diffusion and transport in porous media with variable porosity. Supported by the Hungarian NFSR under grant OTKA T 049819.