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.     

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.     

Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions

Combinatorial proofs are given for certain entries in Ramanujan's lost notebook. Bijections of Sylvester, Franklin, and Wright, and applications of Algorithm Z of Zeilberger are employed. A new bijection, involving the new concept of the parity sequence of a partition, is used to prove one of Ramanujan's fascinating identities for a partial theta function.     

Sixth order mock theta functions

Two new mock theta functions of the sixth order are defined. The main theorem in this paper (Theorem 1.1) provides four transformation formulas relating the new mock theta functions with Ramanujan's mock theta functions of the sixth order. Two further representations of the new mock theta functions are established. Lastly, a hitherto unproved entry from Ramanujan's lost notebook related to sixth order mock theta functions is proved.     

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.     

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.     

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.

     

Partial Theta Functions. I. Beyond the Lost Notebook

It is shown how many of the partial theta function identitiesin Ramanujan's lost notebook can be generalized to infinitefamilies of such identities. Key in our construction is theBailey lemma and a new generalization of the Jacobi triple productidentity. By computing residues around the poles of our identitieswe find a surprising connection between partial theta functionidentities and Garrett–Ismail–Stanton-type extensionsof multisum Rogers–Ramanujan identities. 2000 MathematicsSubject Classification 05A30, 33D15, 33D90.     

The Rogers-Ramanujan continued fraction and a new Eisenstein series identity

With two elementary trigonometric sums and the Jacobi theta function θ₁, we provide a new proof of two Ramanujan's identities for the Rogers-Ramanujan continued fraction in his lost notebook. We further derive a new Eisenstein series identity associated with the Rogers-Ramanujan continued fraction.     

Families of multisums as mock theta functions

In view of the Bailey lemma and the relations between Hecke-type sums and Appell–Lerch sums given by Hickerson and Mortenson, we find that many Bailey pairs given by Slater can be used to deduce mock theta functions. Therefore, by constructing generalized Bailey pairs with more parameters, we derive some new families of mock theta functions. Meanwhile, some identities between new mock theta functions and classical ones are established. Furthermore, based on the proofs of the main theorems, many q-hypergeometric transformations are obtained.     

两个新的mock theta双重和

最近, 张和李使用一个Bailey对, 获得了五个与$q$-超几何双重和相关的mock theta 函数. 本文使用此Bailey对, 我们进一步地建立了两个新的与Appell-Lerch和及theta级数相关的mock theta双重和. 进而也获得了其中一个新的mock theta函数和经典mock theta函数之间的一些关系式.     

Combinatorial Proofs of Identities in Ramanujan’s Lost Notebook Associated with the Rogers-Fine Identity and False Theta Functions

Ramanujans lost notebook contains several identities arising from the Rogers-Fine identity and/or Rogers false theta functions. Combinatorial proofs for many of these identities are given.AMS Subject Classification: 05A17, 11P81.     

Identities Involving Partial Mock Theta Functions of Order Five

In this paper we have given transformations for the partial mock theta functions of order five and also some identities between these partial mock theta functions analogous to the identities given by Ramanujan.     

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.     

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.     

Some integrals in Ramanujan's lost notebook

On page 199 in his lost notebook, Ramanujan recorded three unusual integral formulas. Two of his assertions are correct, and proofs are given here. His third claim is incorrect, but it is true in a limiting form. The integrals arise from a claim in Ramanujan's ordinary notebooks on the equality of certain integrals and series in which the integrand and the summands are themselves the same function.

     

The Bailey transform and false theta functions

An empirical exploration of five of Ramanujan's intriguing false theta function identities leads to unexpected instances of Bailey's transform which, in turn, lead to many new identities for both false and partial theta functions. 2000 Mathematics Subject Classification Primary—33D15, 11P83 Partially supported by National Science Foundation Grant DMS 0457003. Supported by the Australian Research Council.