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 ^2ir (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 + 1 and –1/, where q = e ⁱ. 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.     

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.     

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.     

Identities for Certain Products of Theta Functions

We derive general formulas for certain products of theta functions. Several known theta function identities follow immediately from our formulas.     

On the Products of Three Theta Functions

In this paper, we re-examine several beautiful identities of Ramanujan from the perspective of the products of three theta functions.     

Some Lambert Series Expansions of Products of Theta Functions

Let q be a complex number satisfying |q| < 1. The theta function (q) is defined by (q) = . Ramanujan has given a number of Lambert series expansions such as

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.     

Cubic Identities for Theta Series in Three Variables

We consider three-variable analogues of the theta series of Borwein and Borwein. We prove various identities involving these theta series including a generalization of the cubic identity of Borwein and Borwein.     

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.     

Algebraic Dilogarithm Identities

The Rogers L-function satisfies the functional equation .From this we derive several other such equations, including Euler's identity L(x)+L(1-x)=L(1) and various identities arising from summation and transformation formulas for basic hypergeometric series. We also obtain some new equations of the form where is algebraic and the c _k are integers.     

Partial theta function identities from Wang and Ma's conjecture

Recently, Wang and Ma proposed a conjecture, embedding the Andrews–Warnaar partial theta function identity in an infinite family of such identities. In this paper we use q-series methods to give a proof of the Wang–Ma conjecture. We also present a result which may be regarded as the inverse of the Wang–Ma conjecture.     

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.     

The Borweins' Cubic Theta Function Identity and Some Cubic Modular Identities of Ramanujan

In this paper the author will give new proofs of the Borweins' cubic theta function identity and a related identity relying on the properties of elliptic functions and the technique of comparing constant terms.     

Circular Summations of Theta Functions in Ramanujan's Lost Notebook

Let f(a, b) denote Ramanujan's symmetric theta function. In his Lost Notebook, Ramanujan claimed that the circular summation of n-th powers of f satisfies a factorization of the form f(a, b)F(ab). He listed elegant identities for n = 2, 3, 4, 5 and 7. We present alternative proofs of his claims.     

基本超几何级数的变换公式及Rogers-Amanujan恒等式

本文通过组合反演技巧和级数重组的方法,得到了两个基本超几何级数的变换公式,其中一个的特殊情况包含了著名的Rogers-amanujan恒等式.     

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.     

A Measure of Simultaneous Approximation for Quasi-Modular Functions

We improve on a recent result of Saradha giving a transcendence measure for the quotient of a period of an elliptic curve defined over by its associated quasi-period. In an (almost successful) attempt to include in a single measure both this result and that obtained by Reyssat in 1980, we blend into the modular method ideas related to modular but also hypergeometric functions, as appearing e.g. in André's work, as well as some Galois considerations.     

Modularity and Total Ellipticity of Some Multiple Series of Hypergeometric Type

We collect some new evidence for the validity of the conjecture that every totally elliptic hypergeometric series is modular invariant and briefly discuss a generalization of such series to Riemann surfaces of arbitrary genus.     

On Finite Forms of Certain Bilateral Basic Hypergeometric Series and Their Applications

In this paper we derive finite forms of the summation formulas for bilateral basic hypergeometric series 3ψ3,4ψ4 and 5ψ5.We therefrom obtain the summation formulae obtained recently by Wenchang CHU and Xiaoxia WANG.As applications of these summation formulae,we deduce the well-known Jacobi's two and four square theorems,a formula for the number of representations of an integer n as sum of four triangular numbers and some theta function identities.     

Probability Measures on Dual Objects to Compact Symmetric Spaces and Hypergeometric Identities

We derive, in several different ways, combinatorial identities which are multidimensional analogs of classical Dougall's formula for a bilateral hypergeometric series of the type ₂H₂. These identities have a representation-theoretic meaning. They make it possible to construct concrete examples of spherical functions on inductive limits of symmetric spaces. These spherical functions are of interest to harmonic analysis.