A three-term theta function identity and its applications

In this paper, we establish a three-term theta function identity using the complex variable theory of elliptic functions. This simple identity in form turns out to be quite useful and it is a common origin of many important theta function identities. From which the quintuple product identity and one general theta function identity related to the modular equations of the fifth order and many other interesting theta function identities are derived. We also give a new proof of the addition theorem for the Weierstrass elliptic function ℘. An identity involving the products of four theta functions is given and from which one theta function identity by McCullough and Shen is derived. The quintuple product identity is used to derive some Eisenstein series identities found in Ramanujan's lost notebook and our approach is different from that of Berndt and Yee. The proofs are self contained and elementary.     

On a new circular summation of theta functions

In this paper, we prove a new formula for circular summation of theta functions, which greatly extends Ramanujan's circular summation of theta functions and a very recent result of Zeng. Some applications of this circular summation formula are given. Also, an imaginary transformation for multiple theta functions is derived.     

Eisenstein Series in Ramanujan's Lost Notebook

In his lost notebook, Ramanujan stated without proofs several beautifulidentities for the three classsical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.     

Tenth order mock theta functions in Ramanujan's lost notebook (IV)

Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan's tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan's tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell's transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities.

     

Tenth order mock theta functions in Ramanujan's lost notebook III

Ramanujan's lost notebook contains many results on mock thetafunctions. In particular, the lost notebook contains eight identitiesfor tenth order mock theta functions. Previously, the authorproved six of the eight tenth order mock theta function identities.It is the purpose of this paper to prove the fifth and sixthidentities of Ramanujan's tenth order mock theta functions.The properties of modular forms are used for the proofs of thetafunction identities.     

Cubic Identities of Theta Functions

Many remarkable cubic theorems involving theta functions can be found in Ramanujan's Lost Notebook. Using addition formulas, the Jacobi triple product identity and the quintuple product identity, we establish several theorems to prove Ramanujan's cubic identities.     

Six-variable generalization of Ramanujan's reciprocity theorem and its variants

By virtue of Shukla's well-known bilateral summation formula and Watson's transfor-mation formula, we extend the four-variable generalization of Ramanujan's reciprocity theorem due to Andrews to a six-variable one. Some novel variants of Ramanujan's reciprocity theorem and q-series identities are presented.     

An elementary proof of Ramanujan’s circular summation formula and its generalizations

In this paper, we give a completely elementary proof of Ramanujan’s circular summation formula of theta functions and its generalizations given by S.H. Chan and Z.-G. Liu, who used the theory of elliptic functions. In contrast to all other proofs, our proofs are elementary. An application of this summation formula is given.     

Generalizations of Ramanujan's reciprocity theorem and their applications

First, we briefly survey Ramanujan's reciprocity theorem fora certain q-series related to partial theta functions and givea new proof of the theorem. Next, we derive generalizationsof the reciprocity theorem that are also generalizations ofthe ₁₁ summation formula and Jacobi triple product identityand show that these reciprocity theorems lead to generalizationsof the quintuple product identity, as well. Last, we presentsome applications of the generalized reciprocity theorems andproduct identities, including new representations for generatingfunctions for sums of six squares and those for overpartitions.     

On Eisenstein series and

In this paper, we derive some new identities satisfied by the series using Ramanujan's identities for , and . Our work is motivated by an attempt to develop a theory of elliptic functions to the septic base.

     

Weighted forms of Euler's theorem

In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan's “lost” notebook, we obtain weighted forms of Euler's theorem on partitions with odd parts and distinct parts. This work is inspired by the insight of Andrews on the connection between Ramanujan's identities and Euler's theorem. Our combinatorial formulations of Ramanujan's identities rely on the notion of rooted partitions. Pak's iterated Dyson's map and Sylvester's fish-hook bijection are the main ingredients in the weighted forms of Euler's theorem.     

New analogues of Ramanujan's partition identities

We establish several new analogues of Ramanujan's exact partition identities using the theory of modular functions.     

Four identities related to third order mock theta functions in Ramanujan''s lost notebook

We prove, for the first time, a series of four related identities from Ramanujan's lost notebook. The identities are connected with third order mock theta functions.     

Eulerian series as modular forms

In 1988, Hickerson proved the celebrated ``mock theta conjectures' in a collection of ten identities from Ramanujan's ``lost notebook' which express certain modular forms as linear combinations of mock theta functions. In the context of Maass forms, these identities arise from the peculiar phenomenon that two different harmonic Maass forms may have the same non-holomorphic parts. Using this perspective, we construct several infinite families of modular forms which are differences of mock theta functions.

     

An addition formula for the Jacobian theta function and its applications

In this paper, we prove an addition formula for the Jacobian theta function using the theory of elliptic functions. It turns out to be a fundamental identity in the theory of theta functions and elliptic function, and unifies many important results about theta functions and elliptic functions. From this identity we can derive the Ramanujan cubic theta function identity, Winquist's identity, a theta function identities with five parameters, and many other interesting theta function identities; and all of which are as striking as Winquist's identity. This identity allows us to give a new proof of the addition formula for the Weierstrass sigma function. A new identity about the Ramanujan cubic elliptic function is given. The proofs are self contained and elementary.     

Residue Theorem and Theta Function Identities

In this paper we will use the residue theorem of elliptic functions to prove some theta function identities of Ramanujan. We also derive some new identities by this method.     

On Some Eisenstein Series Identities Associated with Borwein’s Cubic Theta Functions

In this paper, we obtain certain new Eisenstein series identities of level 3. Some of these identities were proved by Liu [4] using the theory of elliptic functions and also by Xia and Yao [7] proved these identities using computer.     

New Modular Relations for the Göllnitz-Gordon Functions

We attempt to obtain new modular relations for the Göllnitz-Gordon functions by techniques which have been used by L. J. Rogers, G. N. Watson, and D. Bressoud to prove some of Ramanujan's 40 identities. Also, we give new proofs for some modular relations for the Göllnitz-Gordon functions which have been previously established by using results from L. Rogers and D. Bressoud. Finally, we give applications of those new modular relations to the theory of partitions.     

Inequalities for the generalized elliptic integrals and modular functions

In this paper, the authors study monotonicity and convexity of the generalized elliptic integrals and certain combinations of these special functions, such as ma(r) and μa(r). Making use of these results, the authors obtain some sharp inequalities for the so-called Ramanujan's generalized modular functions.