Order Mock Theta Functions Research Articles

Second Order Mock Theta Functions

AbstractIn his last letter to Hardy, Ramanujan defined 17 functions F(q), where |q| < 1. He called them mock theta functions, because as q radially approaches any point e2πir (r rational), there is a theta function Fr(q) with F(q) − Fr(q) = O(1). In this paper we establish the relationship between two families of mock theta functions.

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

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 six of the eight tenth order mock theta function identities. It is the purpose of this paper to prove the fifth and sixth identities of Ramanujan's tenth order mock theta functions. The properties of modular forms are used for the proofs of theta function identities.

Transformation Formula of the “Second” Order Mock Theta Function

We give a transformation formula for the “second order” mock theta function $$D_5(q)= \sum_{n=0}^\infty \frac{(-q)_n}{(q; q^2)_{n+1}}\,q^n$$ which was recently proposed in connection with the quantum invariant for the Seifert manifold

Hecke Modular Form Expansions for Eighth Order Mock Theta Functions

Hecke Modular Form Expansions for Eighth Order Mock Theta Functions

A COMPREHENSIVE STUDY OF SECOND ORDER MOCK THETA FUNCTIONS

We consider the second order mock theta functions defined by McIntosh and define generalized functions. We give integral representation and multibasic expansion of these functions. We also show that they are <TEX>$F_q$</TEX>-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.

Modular Transformations of Ramanujan’s Fifth and Seventh Order Mock Theta Functions

In his last letter to Hardy, Ramanujan defined 17 functions F(q), where |q| < 1. He called them mock theta functions, because as q radially approaches any point e 2πir (r rational), there is a theta function F r (q) with F(q) − F r (q) = O(1). In this paper we obtain the transformations of Ramanujan’s fifth and seventh order mock theta functions under the modular group generators τ → τ + 1 and τ → −1/τ, where q = e πiτ . The transformation formulas are more complex than those of ordinary theta functions. A definition of the order of a mock theta function is also given.

Certain continued fraction representaions for functions associated with mock theta functions of order three

Certain continued fraction representaions for functions associated with mock theta functions of order three

Asymptotic behaviour of Ramanujan's tenth order mock theta functions.

Asymptotic behaviour of Ramanujan's tenth order mock theta functions.

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 II

Tenth Order Mock Theta Functions in Ramanujan's Lost Notebook II

Some Eighth Order Mock Theta Functions

A method is developed for obtaining Ramanujan's mock theta functions from ordinary theta functions by performing certain operations on their q-series expansions. The method is then used to construct several new mock theta functions, including the first ones of eighth order. Summation and transformation formulae for basic hypergeometric series are used to prove that the new functions actually have the mock theta property. The modular transformation formulae for these functions are obtained.

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.

On Diophantine Approximations of Mock Theta Functions of Third Order

The paper gives bounds for the approximation of the values of Ramanujan's Mock Theta functions of third order and more generally of some q-hypergeometric functions by the elements of an algebraic number field. Simultaneous approximations for the values of q-exponential function are also obtained. All the results are given both in the archimedean and p-adic case.

Tenth order mock theta functions in Ramanujan's Lost Notebook

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.

On partial sums of mock theta functions of order three

The object of this paper is to define and study the properties of partial mock theta functions of order three, on the lines Ramanujan had studied partial θ-functions. These new partial functions have been expressed in terms of basic hypergeometric function2Φ1. Their continued fractions representations have also been given.

On the rank and the crank modulo $4$ and $8$

In this paper we prove some identities, conjectured by Lewis, for the rank and crank of partitions concerning the modulo 4 and 8. These identities are similar to Dyson's identities for the rank modulo 5 and 7 which give a combinatorial interpretation to Ramanujan's partition congruences. For this, we use multisection of series and some of the results that Watson established for the third order mock theta functions.

On certain Ramanujan's mock theta functions

Ramanujan's last gift to the mathematicians was his ingeneous discovery of the mock theta functions of order three, five and seven. Recently, Andrews and Hickerson found a set of seven more functions in Ramanujan's Lost Note Book and formally labelled them as mock theta functions of order six. In this paper the complete forms of these functions have been studied and connected with the bilateral basic hypergeometric series2Ψ2. Several other interesting properties and transformations have also been studied.

On some relations for the rank moduli 9 and 12

In this paper we prove some identities, conjectured by Lewis, concerning the rank moduli 9 and 12, which are similar to Dyson's identities for the rank moduli 5 and 7 which give a combinatorial explanation to Ramanujan's partition congruences. For this we use multisection of series and some identities for the third and sixth order Mock theta functions, in such a way that all the identities for a given modulus reduce to a single theta identity.

Ramanujan's “lost” notebook VII: The sixth order mock theta functions

Ramanujan's “lost” notebook VII: The sixth order mock theta functions